He is 1 month and 18 days old and he stinks but I don't know when I can give him a shower.

Your puppy is old enough now, just be gentel avoid getting water in the ears or eyes. Use an extra gentel shampoo, I recommend johnsons baby shampoo until your puppy is about 4 months old

He is only 6 weeks old, where is the mother? She should be taking care of the cleaning at this stage.

If the mother was injured or unable to care for the pup then just wash in a mild shampoo and dry thoroughly. It's a good idea to get a pup used to being bathed and groomed from a young age to avoid behavior problems when they get bigger (and harder to deal with).