Hey people - I think I need to replace some electronics on my guitar, I have a problem and I think it is the pickup.

It is an Epiphone SG g400. If I plug my guitar in after a while without using it, the neck pickup is reeeaally quiet sounding.
It's sorta weird because if I play for a few minutes on the broken pickup, it eventually sorts itself out and starts playing again. But does occasionally cut out again too.

No problems on the bridge pickup.

I don't really wanna spend money getting it checked at the guitar store, so if anybody could help me diagnose the problem here, that would be awesome.
Sounds like a bad wiring connection. Pull off the cavity cover and start pokin around the wiring for the neck pickup.
Try spraying contact cleaner (I used DeOxit) on the pickup selector switch and then engaging the switch several times. I had a similar issue (pickup not working, partial volume, scratchy as all hell) on an old Epiphone SG and it fixed it right up.
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I had a problem like this on my Flying V. Couldn't find anything wrong, took it to the shop and it turned out to be a dirty pot.