I stayed at the Goldener Hirsch during the Salzburg Festival last year after it had come off its total renovation. In a nutshell: while the property itself is characterful and beautifully restored, I thought their service was supercilious and distant, and didn't enjoy it at all. The location is good if you want to be in the middle of the city, but Getreidegasse was truly - as mentioned by @
Duck1981 - painfully overrun with tourists and always busy. We drove by the Schloss Fuschl and I'd definitely stay there next time given its spectacular setting, but I would also recommend the Sheraton. It's across the Salzach River but still a very short walk to the Altstadt, has been decently renovated and seemed more down-to-earth than the haughty Goldener Hirsch.
I adore Austria. You'll enjoy it, even in the depths of winter.
khabah