There is no reason to wait two months to get him neutered, he can be neutered now.

They just need time to get accustomed to all the new things, and work out their relationship. I recommend you add at least one more litter box. Ideally you should have three.

You should not let any cat outside for at least six weeks after bringing them to a new home. Give them time to get used to their home and environment before sending them out the door.

Better yet, keep them inside.