My thoughts on questions...

First: Is lack of lubrication the problem?

What type of lube do you use...? Crosman says to use Pellgun oil on their pneumatic pump guns... I personally use 30 weight NON DETERGENT motor oil on mine because it's way cheaper..

Do you already have a lot of lube in the pump cylinder? If so, I don't think that's the problem...

There may be some foreign matter that got into the pump tube... Does it just get hot or do you feel grinding during the pump stroke...?

Second: What is the proper way to lubricate the pump cylinder and piston?

You would first open up the pump lever... Look at the slot on the pump tube... With the pump lever all the way open, you should see a black rubber pump cup at the bottom attached to the plastic spacer... Put 2 drops of Pellgun oil on it. While you are at it, put on drop on each of the moving connecting points...

If you do plan to take your gun apart to clean or look for what's wrong, you will need a roll pin punch, a mallet, and pieces of wood to get enough space to punch the roll pin out... The roll pin is that cross pin on the pump tube in front of the foregrip..