Maybe the lady at the shelter just didn't know what she was talking about, or meant to say 2 months... I would trust her records more than a random lady. A 4 week old puppy would not yet be weaned, would barely have teeth, and would look much like a furry alien.
In most states it's against the law to sell a puppy before 8 weeks, and especially shelters!
I loved having an exercise pen for Fozzie as a puppy. I placed his crate, water/food, a puppy pad, and SAFE toys/chewies in there when I had to be gone for longer than a few hours. He almost never used the puppy pad, but I would prefer it to the floor anyway. Kongs are an amazing investment! Buy one, and buy some peanut butter, smear a spoonful in the Kong and freeze it and you have a preoccupied puppy for an hour.
Bully sticks are amazing for teething puppies. As are lamb ears, cow tendons, cow tails, and other natural, safe chews (not rawhide).
Maybe most importantly... get into puppy kindergarten classes! I recommend finding an Obedience club or behaviorists/certified trainers in your area with both Obedience and play classes for puppies, to get training basics as well as socialization. Look for classes that are positive reinforcement only. The socialization window is from now until 16 weeks, so your puppy should be meeting 100 various people a week and as many friendly, well-mannered dogs of all ages as possible.