We arrive by Train from Barcelona at 09:48 in Girona before a lunch reservation at 13:00 and are looking for a quick tour with a private guide or just suggestions on how to get around by ourselves to see the cathedral and possibly do some shopping.

thanks.

I only know personally private guides who operate in Girona from Barcelona, but your most important first stop when arriving should be the very helpful tourist office, located at Rambla de la Libertat 1. The people there are very helpful.
https://www.girona.cat/turisme/eng/index.php.

You might contact this one (can't vouch for her because I don't know her personally)
https://guiagirona.es/en/english-2/

We are going to Girona in September and I've been researching the guides for the town and the area. Here is one who might be good, his web site says he also does custom tours:https://www.toursbylocals.com/Besaluprivatetour