Welcome to the Cat Forum, I'm sure you'll get lots of help here.
Sounds like this is his first day with you...it will take time for him to adjust to be away from mom & littermates, so not eating too much is pretty normal. Sleeping in the litterbox is a security thing. Does he have a bed or give him a cardboard box with some towels as a lining...basically provide him with a snug nest in an out of the way place.
As Tim said, litterbox & food dishes should not be near each other. If he's closer to 3 months and you have a house with more than one level you may need to put more than one box as he may not realize that he has to 'travel' to get to the box. If he's 4 months, he should OK.
I don't think you should let him out to see Mom. I would recommend making him an indoor cat. Please encourage your neighbor to get Mom spayed so she doesn't have more kittens they don't want. And of course get your little guy neutered no later than 6 months.
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