Hi, Christina! Welcome to the forum!!
Showing can be as low-key and relaxed to as expensive and involved as you want to go. The best way to get your feet wet would probably be to look up a local open show or schooling show. It depends, but the classes here are usually cheaper ($10-15 per class plus number/grounds fee, usually another $5-10, is about the norm in my area, once you get out of 4-H/youth shows) and the competition is pretty local. Sometimes these kinds of shows are hosted by boarding/training barns, riding clubs, etc. 4-H clubs will sometimes host open shows as fundraisers. These sort of events can be a great way to get started and get a feel for things in a smaller-scale show atmosphere.
If you're looking to get into the Arab breed show game, your best starting point would most likely be to contact the registry association and investigate what exactly is available and how to get involved. I'm not a lot of help there... I've never shown a horse in a breed-specific show. My understanding is, though, that the cost of competing is somewhat more (you're paying for an association-approved judge, etc.), and depending on where you are located you may need to do some traveling to get there. One of my old riding instructors pretty routinely trailers her horses hundreds of miles for AQHA shows.
I find horse showing, especially Halter classes, to be very much like showing dogs. The same basic process - the judge has a standard for the breed/type, and compares your horse to that standard. We don't whittle down breed by breed, and then the best of the group, and best of show, though. There are divisions according to breed/type (I.e., Arab Stallions, Arab Mares, QH Mares) that are placed, and that's it for the class. There are often high-point awards, though, that are given to the highest average score across a division, that sort of thing, at the end of the show.
If you are interested in seriously campaigning your horse on the breed circuit, I highly recommend enlisting a coach or trainer to help you. Most any sound horse can be shown successfully at some level, but there comes a point when, to be competitive as an amateur, you need to have a pit crew who knows the breed and the discipline on your "team" to back you up and help you with the finer points. Perhaps not quite so important with in-hand classes as under-saddle classes, but it is still worth a clinic or a couple of lessons with an experienced Arab Halter trainer/exhibitor. That can add to the overall cost, but there are a lot of owner/trainer/exhibitors out there - it all depends on your own experience in training and showing horses.
Again, I'm not too familiar with breed shows, but in terms of open shows 6 years is far from too old to show. Halter classes typically aren't divided by age beyond 4 years, and unless any other class has an age range attached to it, you can show your horse for as long as he is sound and healthy and you both enjoy it!
The only issue that I can see from your OP is that your horse is a stallion. While not so much a problem in breed-specific shows, a lot of smaller open shows have liability issues with stallions. Some show grounds require specific measures be taken in handling/containing stallions, and some simply don't allow them. That depends on the event - you can find that information out by contacting whoever is running the show.
Good luck, and have fun!!