I, and many others, have had great success with Auto-Rx. It's an engine oil-carried cleaner, and it rocks. I highly suggest running it.
http://www.auto-rx.com/index.html
For H22 motors, I had good success running the high-mileage labeled regular oils, because the motor burned a lot less oil those times. I ran Amsoil synthetic for nearly all the car's life, and the H22 definitely runs great on that. Be aware though, that most synthetic oils tend to burn faster in the H22, so you have to be a lot more on top of keeping the oil level up.