How can I connect a 120V compressor to a 3 phase WYE 208 electrical system?
In a 208Y/120 4 wire Wye system, you'll get 120 volts between either ungrounded (hot) conductor and the grounded (neutral) conductor.  So if you're trying to connect a 120V appliance, all you need is one ungrounded (hot) conductor, one grounded (neutral) conductor, and one equipment grounding conductor.

The ungrounded (hot) conductor will connect to the brass colored screw terminal on devices (the small hole/blade), while the grounded (neutral) conductor will connect to the silver colored terminal (the larger hole/blade).  The neutral conductor should never be connected to the equipment grounding conductor, or equipment grounding terminal on devices.  

Notes:
Since it sounds like you're a bit over your head, you should probably contact a local licensed Electrician to finish the project and for any future electrical projects.

While you can solder connections in house wiring, it's not a common practice anymore.  You'll also want to insure that you are using solder that is rated and labeled for the use, and that your solder joints are strong and insulated.