I think my cat is unwell

Question
I got a cat from the RSPCA in November and he has settled down really well in the 2 months that we've had him. He's always either used the litter tray or gone outside when he's needed the toilet but this morning has started peeing around the house. It doesn't appear to be deliberate. He will go to the litter tray and spend about 3-4 minutes doing a tiny amount of pee and then on his way back downstairs from the litter tray will pause on the stairs or at the bottom of the stairs  and crouch as if peeing. Again this may take a few minutes and then we find a small puddle on the floor. He has never peed in the house before and we are worried there is something wrong with him.

Answer
You are right, there is something very seriously wrong with him and he is trying to tell you that.
He most likely has a case of cystitis, which is inflammation of the bladder. However, in a MALE cat, this can quickly turn into a deadly blockage of the urethra, the tube that is in the penis that carries the urine out of the bladder.
The bladder fills up with a toothpaste-like sludge full of dead white blood cells, bacteria and crystals.
Then it plugs up the urethra and your cat can't urinate at all. Usually you see the beginning, with small amounts of urine and the urge to go all the time.
This can be very life threatening due to the build-up of toxins in the blood, so you need to get him into a vet NOW.
Even 24 hours can make the difference between him making it and not making it.
Even if this mean going into an emergency vet, you need to act quickly. If this was a female I would tell you that you could wait until Monday morning, but since it's a male, and I have seen many die from this and from owners waiting, please get him into the vets asap!