Caps are a good place to start. If it was mine and knowing what I know as outlined below, I'd try and replace only the problem components for a start -look at Rhino's suggestion above, but that said... I have read tales of sudden destructive failures in Omni's, so maybe a full redo is the best approach. PM me for manual PDF's if you want.
Knowing what I know: I replaced ALL the seventy-some-odd caps in an Omni 2 a few years ago. It also had nearly frozen sliders so I did the Simple Green board soak* thing too, and it worked perfectly after. Really, 100% perfectly! Still does.
I felt empowered so I bought a barely functional Omni 1 and did the same -it remained barely functional after. I had a synth tech work on it and now it's about 90% functional.
Replacing all the caps is about a days work if you know what you are doing and are careful. Will it solve all the problems? Maybe. There are threads here and blog posts and etc where you can get more pointers.
Is it worth trying? Absolutely. For a simple string and synth machine it's really good sounding.
*Simple Green board soak is: you remove the boards with sliders, put them in your shower, spray them down with simple green -especially in the sliders, let them soak for a few minutes then use the shower head pressure to clean them. Lots of c**p comes out. You can find Youtubes of this for other Arp's with the same sliders. Once they are clean you rinse them in distilled water -I did anyway, to avoid hard water deposits then put them in the sun to dry. Try and time it so the sun is shining right when they come out of the rinse. You can also use a spray bottle of rubbing alcohol to 'chase' the water away. Apply fader lube and reassemble when perfectly dry. You might want to do this bath before you recap...
Hope this wasn't too much info...