How do I fix a stuck piano key on a upright home piano

If you cannot see right away something stuck between a keyboard rail and a piano key, your piano needs to be adjusted by a proper piano technician. There are too many moving parts where 1mm makes all the difference. Down in Florida, our humidity is pretty high, and when we bought our piano, there were 3 sticky keys. Since that was a high end upright Sauter piano, I called a technician from www.europianosnaples.com. The guy came out, took apart the keyboard, nothing wrong there, all the key bushings (guess that's what they are called) are fine, keys were not touching the rail. Tech found a few weak springs in the action (I was explained that they push back they key). Took 3 hours to replace them since they were in a place, where I wouldn't dare to get into. http://www.europianosnaples.com/

You'll likely need to try removing the keys, so you can see if something is stuck underneath it. That would be my first move.
In order to do this, follow these steps:

The front panel and the keyboard fall are designed to be easily removed for tuning purposes anyway.

To check it out....The front panel likely has a little clip either side near the top on the inside. Just undo these clips and lift the panel out. The keyboard fall should then be easy to lift out too.

There may be a long wooden rail which you need to take out also. Once inside, the keys themselves can be carefully lifted off the central spikes on which they sit.

Remember how you did this so you can get them back in again.

Since your piano is so old and worn, Often times, the wooden rail in front of the keys (just above your knees when you are playing) bows inwards a little due to age and temperature/humidity fluctuations. This then causes the little white front bit of the keys to jam against the wood and you have to physically lift the keys back up to return them.
If this is the problem you will need to take out the keys and shave some wood off the inside of this rail to allow the keys to move up and down freely again.
Or sometimes you may find you have a small object caught under a key or two. Much simpler to remove and free up your keys, if this is your problem..
Hopefully, this is all you will find wrong with your piano.
If all else fails...
I wouldn't attempt to tune your piano by yourself. I would seek out an experienced piano tuner for that.
But it is worth attempting to repair it first before you go that far.
Good luck !!