How can I use one AppleID for Contacts, Calendar etc, but an other for purchases?
On the iPhone, open the App Store. Click the "person" icon on the top right. This will allow you to log out from your new Apple ID, and then login with your old Apple ID.

This will not overwrite any date, nor will it change which Apple ID is used for contacts, calendar, iCloud or anything like that. It will only change which Apple ID is used for purchases on the App Store (including downloading of free apps).