Welcome to the forums and the ever exciting world of budgie ownership!
You have certainly come to the right place as you will find all you need to know on budgie care here.
Since you are new to budgies, the information on the several articles and sticky threads located at the top part of each section of the forums will be extremely useful as you become more familiar with the species.
Be sure to pay extra attention to the Diet and Nutrition and the Taming and Bonding sections of Talk Budgies.
What you have described is considered normal behaviour for a young budgie.
Throughout the day, budgies can have bursts of energy where they are very active and they will also have moments where they are more quiet and want to have shorts naps. If your budgie is eating and drinking, is active and the poops look normal, then all should be well.
The fact that your budgie is wary of you is also perfectly understandable, given the fact you have only had him for 4 days. It takes time for a pet bird to adjust and be fully comfortable and settled in their new home.
Your budgie is still in the early stages of getting to know his new family.
You may find the information on these links useful: