It is all going to vary on the content of your game center. Games, consoles, computers, arcade cabinets, furniture; these are all things that you have to take into account. It is probably going to be cheaper for you to invest in consoles as opposed to computers given that you're likely to provide a more stable gaming experience for your customers through consoles. Computers are great for smooth performance and graphics but they have to be top tier to provide that, and you're going to be sinking in a lot of money for multiple systems that can play the latest titles on max settings at 60 FPS.
Don't forget that unless you have your own property, you're going to have to take rent into account as well. I don't know how it is in your area, but if you want to provide refreshments for your customers as they game, such as drinks and snacks, you're going to need a license for that as well.
We can't really get into cost estimates until you've figured out what exactly you want to provide, but it is definitely going to be a pretty penny. Keep us up to date, I can't wait to see how this goes!