HELP we accidentally took a cat home 😭

This evening, when my partner and I were walking home, a tuxedo cat leaped out of nowhere and purred at us non-stop. It followed us up the hill and almost into our building complex.

We were worried we accidentally stole a neighbour's cat, so we went back down the hill and knocked on four different neighbours' doors to ask if anyone owned a black cat. They all said no. We went back up to our flat, and the cat followed us inside. It was uncertain at first, so we lured it into our actual flat with some milk.

We tried giving it dried cat treats with catnip (my old neighbour had a cat so I had some left over), but it wasn't interested. We both have never owned a cat before, so we are at a loss. We are thinking of going to a vet tomorrow to get it checked for a microchip.

What should we do right now? It looks very young. What should we do if it needs the bathroom? We don't have a cat litter box. Helppppp us please!