The cheap eBay solenoids are not the same as the ones you find in full auto arcade recoil guns. That's not to say you can't find good solenoids from non-arcade stores. You definitely can but they are rarely cheap. I spent an ungodly amount of time on this issue. I wasted a lot of money on cheap solenoids before I learnt that lesson.
If you don't need full auto recoil then you can get away with a cheaper solenoid like the one found in the Ultimarc recoil kit. It will still set you back $30 though. Plus, the main issue you'll have with putting one in a gun not specifically built for it is making it fit. I would recommend saving yourself time and money by finding a deal on one of the PS1 jolt guns (also known as "real Arcade guns"). I got mine for $40 delivered and they come with a decent full auto capable solenoid, PSU and pedal.