What happened:

I turned up the heat on the water heater in our rental house (from "Warm" to the spot between "Warm" and "Hot").

The faucet in the tub spout out some slippery water and some crumbly black bits, but it had a nice heavy flow of hot water. The slime-feel and bits went away. All other faucets in the house function properly as well, except for one faucet in the upstairs bathroom.

Cold water comes out just fine but when I turn the hot water handle, only drips come out.

I cleaned out the aerator and it was fine.

I tried small thumps on the hot water hose and pipe under the sink; turned off the hot water under the sink. I can't seem to dislodge whatever is blocking it.

Is there something I can do myself or do I need to call in a plumber? I really would rather not call in a plumber.

Thanks!

This is what happens when you don't flush your heater and allow minerals to build up in it. The minerals get sucked up in the system and the first faucet that makes a draw gets the crud sucked up into it blocking the inlet ports of the cartridge, in 6his case your lavatory faucet.
The solution: shut the water off and open up the valve. Check and clean the inlet ports on the cartridge. Before you reassemble turn the hot water back on for a moment to flush out the supply.
For long life and fewer troubles you should keep your heater clear of mineral build-up by flushing on a regular schedule. Let me show you how. Attach a hose to the boiler drain at the bottom of the tank. With the pressure on, open the boiler drain and let it run until the water runs clear. You will see a spurt of red,(rust) followed by white grains,(lime or calcium carbonate). This shouldn't take more then a few minutes. Do this monthly to keep it clear. Now flush out your hot water lines on ALL fixtures. Now pull each aerator and clean the screens. Be sure you put them back togather the same way you took them out. Don't forget to flush it out every month. Your heater will thank you for it. Hope this helps, Tom