If you go for a public school you want it to be a Gymnasium. Gesamtschule also offers upper grades but are usually of lower quality (of course there are exceptions but you would need to be a local to know those).
Gymnasium used to go up to grade 13 but most states changed that a few years ago to just going to grade 12. Parents and students complained and so some states have changed back to 13. I believe those states are Bavaria (considered to have the toughest public school system in the country as they have the strictest criteria as to whom they allow to go to Gymnasium),Northrhine-Westphalia and Lower Saxony. Grade 13 is probably more appropriate for a PG year. Please double check this info as it seems to be a situation in flux.
I am not an expert on boarding schools. But these are some of the top and most well known boarding schools in Germany (don’t know about Austria or Switzerland):
Schloss Salem (probably the most famous of German boarding schools)
Louisenlund
Schloss Torgelow
Schloss Neubeuern
Birklehof
Instruction at these will be top notch.