We're from Alaska and must have air conditioning when we travel to Vienna this June. We'd prefer Old Town area, smaller hotels.
Pension Sacher. Wouldn't stay anywhere else.
I'm not sure when you're going in June, but I was there in early June, and it was flippin' cold...in the low 40s and actually spitting a few flakes of snow one morning. You never know about weather. Anyway, you'll have to check if this place has AC: Pension Suzanne, located ½ block from the opera house. Used to have 4.5 stars on tripadvisor...probably still does. Great small hotel...even sort of fancy but priced reasonably.