How to solder the 4-pin battery connector back to phone's motherboard?
It's hard to say, if you ripped up the pads (which looks like one of them is ripped up) then you'll have a problem. The copper needs to be continuous (especially in high amperage connections). This is even with soldering.


  Is there a way to solder it back? 


First off the connection must be soldered. Secondly, you must create a low resistive pathway that can handle the current, this can be done by blue wiring. If you don't know how to calculate the current or blue wire, then find someone who does. 

Blue wiring involves soldering directly to the pad of the connector, then finding where the trace goes (if you have room) and either peeling back the solder mask, and/or soldering to another component on the same trace. This may not be easy on a device with limited space. 


  I'm gonna try to just attach it and see if 2 Phillips bolts make enough
  pressure to make contact for the 4 pins to touch the mobo. I can
  certainly superglue it back.


This will most likely result in failure, with modern phones, charging can use amps of current. If a proper connection isn't made, the battery will either not charge, or will heat up a resistive connection (think fire/melting).