No you cannot pick up a package from USPS distribution centers.
When your package arrives at or is at any USPS distribution centers the package is going through the sorting process and is being scanned, sent down conveyor belts etc.
The USPS sorting facility or distribution center would never be able to quickly find your package to hand it to you if it were possible.
So when you're having a package being shipped through USPS and it is at a USPS distribution center even really close to your address you'll need to wait.
The USPS package should arrive within 3 to 7 days depending on how fast the shipping is.
Most packages through USPS have been delivered to my post office box within 3 days.