Hey Julia, the German rules say that a negative test is only accepted when it has been taken up to 48h upon arrival in Germany. But the results don’t need to be there on arrival.
Alternatively you can take a test after arrival and quarantine until the result of this test is there.
Depending on where he will arrive in Germany you can do that right at the airport.
My partner landed in Frankfurt and we pre-registered for a test and walked straight in, got the test and had the results 6h later.
The test is not necessary to enter Germany, for that just the declaration form for your relationship and the other documents are necessary.
The PCR test is just to avoid to quarantine for 10 days.
For us everything went smooth in Frankfurt. The boarder control only asked for the invitation letter, it took 2min and the guy was very nice.
To be sure about what to do, call the “Gesundheitsamt” of your town.