I have obtained a rather cheap P-bass imitation bass and, while it plays well in the lower register, about the 9th fret the action starts to rise and after the 12th fret it's in to the "wow, this actually getting harder to fret" territory.
How do I approach troubleshooting why the action is high? How do I go about adjusting the bass to lower the action?
Definitely check that the neck is not bowed first. If it is, truss-rod adjustments can fix a lot of problems in limited ways. Too much of a bow, and no amount of truss-rod adjustment will fix it. If everything looks fine there, then you have two things to consider, action wise: height of the nut, and height of the bridge.
If your instrument plays fine at the first position, but the action is too high above the twelfth fret, then just lowering the action at the bridge might not be enough. It could make the action too low at the head. You would then have to raise the string height at the nut. this would require a new nut, as the slots cut into the current nut are too.
Also, after adjusting the action, you will need to check the intonation to make sure the string length is still right for the fret spacing. (even a millimeter off can make playing in the higher positions sound out of tune compared to the lower positions).
If you feel confident that you can make these adjustments, then take a stab. You can also find a reputable repair shop in your area and take it there, but if all you want is a playable bass, you might end up spending more time/money on repairs and adjustments that the bass is actually worth, depending on what is wrong.
First of all, I would check whether the neck is straight.
Place a ruler against the frets (the edge of the ruler against the frets) and if there is any space between any of the frets and the ruler, the neck is not entirely straight. If there is no access to any sort of truss rod adjustment, then you will probably have to put up with the high action. If there is a bow in the neck, the action will appear higher at higher frets. If you lower the saddles in order to fix that high action, you will get fret buzz at the lower frets.
If the frets are straight, try adjusting the action at the bridge. Nearly all basses have a setting somewhere on the saddles that allows height adjustment. Once you have them a comfortable height, or the lowest you can without any buzzing, you will need to check, and possibly, fix the intonation.
Tune the guitar to pitch using a tuner. Play the 12th fret harmonic on the string, and then the 12th fret note. In short, the rule is:
If the fretted note is higher (sharper) than the harmonic, then the length of the string needs to be lengthened. If the harmonic note is sharper, then the length of the string needs to be shortened.
The way you adjust string length depends on the type of saddle, but most cheapo basses usually have a Phillips screw of some sort that you can adjust.
There might be a problem with the nut, but I don't know a lot regarding them.
In this case, I would usually recommend the instrument be taken to a guitar tech. They usually do a fantastic job on fixing the instrument's little niggles straight (and your bowed neck if necessary), you will be able to hear and feel the difference when you get it back. Depending on the service, it may be a tad pricey (£40/$64) but as you say the bass is pretty cheap, so it shouldn't hurt too much anyway. You can always fix further issues yourself, but whenever I happen to acquire an instrument I always like to have it perfect and go from there.
Hope this helps :)
It sounds like the neck is not straight. Ensure it is bolted or otherwise attached to the body, and then check up at the nut for an allen wrench slot. Many necks have an adjustable tension (truss) rod. ( see: http://www.tunemybass.com/bass_setup/adjusting_neck_relief.html )
When done improperly, this can break the neck, so...don't break the neck.
A few comments on the above:
Don't be afraid to make a minor truss rod adjustment, BUT... Always make absolutely certain you have the correct size wrench and it is fully seated in the nut. If not, you can easily break the truss rod nut. Never make a single adjustment over 1/8-1/4 turn. After every truss rod adjustment let the instrument "settle-in" a day before making another adjustment.
Capo at the first fret and fret a string at the 15th. The string is a dead-straight line from which you can check for any curvature in the neck.
If the action at the 1st fret is good, saddles are bottomed out, the neck is straight and the action remains ridiculously high you may need to shim the neck. If so, use a true wood shim that is cut to fit the full neck pocket - don't do a half_*ss thing like using a pick, credit card sliver, etc. as a shim. (The reasons for this can be found online, but let's cut to it and say shortcuts can ruin your neck).
Try only as much as you really know how to do. After that, get your instrument to a luthier or competent guitar-tech.
