Does the coil have a diode? If so, when the wires were reversed on the coil, it probably blew out the diode so that coil diode should be replaced first before anything else.

Is that leftmost transistor on the board connected to the C12 blown cap? If so, look in the manual for the solenoid chart for the upkicker and verify it's associated with leftmost transistor. Can't be sure from your pic, but it might be Q1.

Assuming Q1. If upkicker goes to Q1 in solenoid chart, then replace C12 first and test Q1 using a voltmeter in diode test. Look up transistor test method on pinwiki site. Replace Q1 if it tests bad. If Q1 tests good, then power up and verify upkicker works. It's possible C12 and/or fuse might blow again. If that were to happen replace both C12 and Q1 at the same time and test again.

I think you'll find that's a capacitor that has exploded. Pull the board and check the traces on the other side. They may be cooked as well. Also test the transistors associated with the coil. They may need replacing also.