You can definitely fix it. Replace the parts that are broken. If you determine that you need new pickups, you'll get plenty of good recommendations here.
For testing electronics parts it helps to have some equipment, but you don't need to invest in anything expensive or complicated. I made a few circuit testing gizmos out of wire and alligator clips. Basically about a foot of insulated wire with an alligator clip soldered to each end. You can bypass any or all the controls with these - for example, to bypass the pots and jack and just test the pickups, just clip one end of each wire to the pickup wires, and the other end to the guitar cable (the tip is hot, the shaft is ground). Then tap the pickups with something metal. You'll hear it coming through the speakers. Cheap to make and they are useful in all sorts of situations where you want to bypass a pot, switch or whatever.