Whatever You Decide....Use Aramith Cleaners
You can clean your pool balls with just 409 Cleaner. The lustre won't be there on the finish but the pool balls will be clean. To restore the shiny finish, which does improve the roll of the balls on the felt, you can manually polish the pool balls Aramith Cleaner. If the pool balls are really dirty and old, then you should use Aramith Restorer instead.
Personally, I'd look for a automatic machine. I know it's expensive but believe me, the luxury of having something that simple available is just great. I found a Ballstar on eBay for $330 shipped and it was like brand new. It holds 8 pool balls and the entire process (cleaning and polishing cycles) is only 2 minutes. Then I pop in the remaining 8 balls and in another 2 minutes, all the pool balls are cleaned and polished.
Counting the time to dab just a fingertip of polish on each pool ball before loading and also including the unloading/polish application/reloading of the second 8 pool balls, the entire process takes under 10 minutes and is pretty much labor free. And the darn balls look sparkling and you can see a brilliant reflection/image off every single ball. After a few consecutive cleanings, the pool balls remain shiny much longer.
I take my Centennial pool balls to the pool hall when ever I go there. I carry them in a Aramith case that's just a terrific case to transport your pool balls. Even on the relatively dirtier pool hall tables' cloth compared to what you'd otherwise play on your table at home, my pool balls can be used up to 3- 4 visits (2-3 hrs play each visit) before I want to clean them. At home, you could probably play with the pool balls probably 20-30 hrs. before you might want to clean them presuming your table cloth and all the players' hands are kept clean.
It seems like a luxury but it's really a wonderful gadget to own. I had to sell my table but I kept the Centennial pool balls and substituted just a basic set of Aramith pool balls with the sale. I made sure to also keep my Ballstar Automatic Pool Ball Cleaner and frankly, since now I only play in pool halls, I just make it a routine to clean my Centennial pool balls every other time. The pool balls always look brand new and it only takes a few minutes. I do while watching ESPN or something else but it doesn't take much time and it's so simple.....turn the knob and remember to be nearby to listen for the chime sound after the entire cycle (2 mins) is done. Repeat the process one more time and you have brand new looking pool balls.
You can do the very same thing manually which I was doing before getting the automatic cleaner. I used the 409 Cleaner and manually applied and polished the pool balls with Aramith Cleaner. The pool balls came out looking clean but not nearly as shiny as with the machine. And besides, my hands would tire and it took a whole lot longer too......a lot longer. There's also the home version on You Tube that's been mentioned but that's just not my thing......go to the garage, get the bucket and the Mikita drill etc. It admittedly works but for my tastes, that's more of a contraption than what I have in mind for a pool ball cleaner.