Congratulations on getting a new parakeet!
Your bird sounds a lot like my parakeet T-bird was when we first got her.
She stayed at the top of her ladder for the first day or so and looked like a little statue that never moved!
Have you tried some spray millet, that was the first thing that T-bird liked to eat.
We were worried that she wasn't drinking at first so we got a test tube water bottle which she preferred drinking from instead of the one that came with the cage.
Parakeets can be shy at first and not like change or sudden movements or noises.
Make sure you cover the cage at night, it will make the bird feel more secure.
You also might like to whistle quietly and see if your bird responds.
Our bird also liked to talk to the wild birds outside since it was summertime when we brought her home.
Hopefully as your bird gets used to its new surroundings it will eat and drink more and therefore poop more.
You might still just want to call the vet tomorrow to be sure that everything is ok.
Good luck! I hope to see some photos soon.