Will Bioshock run well on this type of CPU: Intel(R) Pentium(R) Dual CPU E2200 @ 2.20GHz?

Yes, it will run fine on that. I am running it on a dual core notebook @2.1 GHz. I think you'll find your video card and your desired resolution to be more limiting.

I can't run it tolerably at 1680x1050 (my native LCD resolution), for instance. Recommended System Requirements: CPU: Intel Core 2 Duo processor System RAM: 2GB Video card: DX9: Direct X 9.0c compliant video card with 512MB RAM (NVIDIA GeForce 7900 GT or better) DX10: NVIDIA GeForce 8600 or better Sound Card: Sound Blaster® X-Fi™ series (Optimized for use with Creative Labs EAX ADVANCED HD 4.0 or EAX ADVANCED HD 5.0 compatible sound cards).

The processor is fine. You didn't specify your RAM. I recommend having at least 2GB of RAM, but having 1.5GB or so should do.

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