A doddle, if you are in J with an EU passport and can check your bag through to VIE. Do not bother with the FCC, just go landside and follow the signs through the tunnel to T1.

OS, along with other LH cronies, now use a separate check-in area for premium pax at T1 (Zone K - formerly BA's beloved Zone R) - which, crucially, has its own separate security screening, so that you bypass the main security jam. It's down at the far end of the terminal from the lifts that bring you up from the tunnel - you have to go outside at the furthest set of doors and through a sort of walkway.

I'd say you have time for a quick shower at the AC Arrivals lounge if your nerves are good. Wait breakfast until you get to the *A departures lounge at T1, which generally has more and better food.