Raiding with horse archers is fairly common now in RoN, it's easy to do and can be effective. However, defence against them is very easy and no intermediate player should be caught out.
Horse Archers are best used separately and individually. A single HA can kill an enemy citizen in two shots but due to RoN's focus fire rules, two HA do not kill a citizen in one shot. It is better to send them to two locations which means the first can be on a mission as the second is being built.
When the HA is produced, set it's stance to raid and way-point it to known and likely targets in your enemies territory based on your scout's finds. It is often useful to allocate a control group number to each HA so that they are easy to locate as you manage your economy.
Standard defence against HA raiding includes a tower to cover vulnerable gather spots and LC units for fast response. Even with these defences, HA raiding can be beneficial, but it's impact is much less and marginal at best. Barracks units are not very good at all, HI die horribly to them, LI don't damage them much. FA are effective but the main problem is that they more far too slowly to intercept.
The idea behind this strategy is to hit the enemy so fast that they can't mount a suitable defence using LC. A fast raid can hit the enemy before the 3 minute mark. If performed well, this raid can force the enemy to bell up and can lock them into ancient age for a very long time.
The goal here is to concentrate on food and timber and get to classical around two minutes with a small economic growth. Research science for cheaper advance and ruin benefits and make three citizens to improve your farms and timber rate. Research Military 1 and then save up for classical. During this wait for classical food you are almost guaranteed to get a wealth rare, don't prevent this. Before hitting classical you need to make a decision on if you want to research Military two in ancient or in classical, the key difference is your use of starting Metal. If you want to research it in ancient then wait until your food is at 280 before hitting classical (225), then as classical is being researched you will get the required food for Military 2 before classical research is completed so you can queue it up paying ancient age costs. As soon as you hit for classical, send a builder forward to the border so you can lay a stable down towards your enemy the instant you reach classical.
You should easily be able to afford 2 HA and possibly a HC, set their stance to raid and send them over to the enemy land as soon as they emerge. Target lumber areas and remote farmers. Keep moving them around but you should be able to place one on each city to keep the enemy economy totally suppressed. Watch out for the construction of towers and barracks or stables, kill the builders as priority, even if there are 4 or 5 builders you should be able to prevent the construction by killing them all, though you will then need to guard this site as it will be easy to complete. A scout can be very valuable in the area but I would recommend that you use him to explore and pick up ruins as your economy now needs priority. Should help arrive or the enemy gets some defence up, it is worth sacrificing your HA to get a couple more citizen kills to get maximum value.
Summary
Science 1, 3 citizens and scout
Military 1
Classical
Build Stable and Military 2 (not Bantu)
2 HA on raid stance and attack immediately
Keep moving an harassing both cities as long as possible
Concentrate on classical economy
Similar to an ancient HI rush, you have sacrificed your early development to attack your enemy. Once your initial raiders are made your main focus should be on economy. Your food situation will not be bad at +60 but your lumber will be bad at +40. Worst still is your wealth, you have none and you are now in classical. Ideally you need to build a second city, a temple and a market. Then you have to get Science 2 as a matter of priority, without this your economic growth is severely hampered due to lack of economy buildings so getting market trading going with caravans and the taxation upgrade is a priority. If other aspects of your economy are working well then you may find it advantageous to hit commerce 2 first so you can sell food for that vital gold.
You can expect between three and six ruins during the build up phase. Mostly you will want food off these ruins, though timber is also useful and these are easy to manipulate with temporary construction of citizens and buildings. However, you will also require at least one wealth ruin to enable a second HA, but more wealth ruins later on are very desirable to enable a classical age economy with merchants and scholars. Unfortunately, ruins in early classical age are likely to yield metal rather than wealth.
This strategy should give you a very significant advantage against an unprepared opponent so you have a number of options on how to proceed. You could build an early attack with siege or just out-boom. One nice cheap option is to build a barracks and trickle HI into a city, send the odd LC in if archer units are made, if you can capture the city you will have a big enough force to quickly finish your crippled opponent.
What makes this strategy so devastating is it's speed, by hitting when you are in ancient you have no defence capability. You do however have a little warning. When your opponent hits classical around two minutes that is your queue to act. If you are able to jump to classical then do so immediately, you have to get a stable down and 1 LC out asap to kill his investment, don't forget you need Mil 1 too. If you are not able to hit classical then you must get Mil 1 and build towers in key locations, two should suffice, but don't underestimate the build time of towers, you only have seconds so get them up with multiple builders and think carefully where you place them.
If you are in a team game and get caught with your pants down then scream for LC help the instant you suspect the HA raid, make sure you get a confirmation, all you need is a single LC so it shouldn't cripple an ally to provide this.
Bantu
Bantu are excellent at this strategy, the fact that they do not need military 2 to build HA saves them a lot and even allows them to throw a cataphract into the raid if they want. Their cheap cities makes economic recovery easier.
Egyptians
Egypt have good chariot units and a single city economy better than any other with their wealth producing farms.
Mongols
Everyone knows about the power of the Mongol raid, with 2 free HA on stable build the Mongol raid can be even faster than expected. Your only problem is that it will be expected, so why disappoint them?
Nubians
Market manipulation and rares gives the Nubian player plenty of options for fast strategies.
Romans
The extra gold income from cities gives Romans a surprising raid capability.
Obviously any land map is good for this but some are better than others. Australian outback will allow you to raid very fast due to the distance being so short, however, the defender will have income from the sea so the raid is not quite as disabling as it could be. Warring states has a similar benefit and problem except in team games, help is likely to arrive earlier. Great lakes is not so good as the distance your HA have to travel to raid is usually much greater, your enemy has a better chance of getting to classical and a stable. Southwest Mesa is surprisingly good, the enemy expansion city will be in easy range and the extra pathing to the enemy capital not so much of a problem as with a rush.