As is so often the case, there is much truth in all the reviews posted; and experiences vary depending on time of year and placement of room.
Overall, Hotel Alexander is a good place for American tourists. The staff, while reserved, is polite and helpful in English. The rooms are small but not tiny. The TV is small and with one English station (CNN). The elevator is standard Euro smallish. The breakfast is the classic Germanic fare: rolls, preserves, cold cuts, cheese, juice, musli, milk, and good coffee. Don't be shy to ask for a soft-boiled egg (weich gecochtes Ei) if you wish. The quality of the breakfast components is high. The breakfast staff knows only a little English, so practice your restaurant German!
Bring a hair dryer (and plug converter) -- the rooms have none. Bring shampoo and wash cloth -- ditto. There is no A/C, so be concerned in hot months.
The location is off the beaten path to the Northwest of the inner city. The quiet neighborhood, however, is near a branch campus of the University, so there are a number of cafes and restaurants nearby. (The student center there has an ATM machine, a few vending machines, etc.)
A key factor in favor of this hotel is that the D trolley stops close by and it goes right down to the main sites in about 15 minutes. After you arrive at the hotel, go to a nearby Tabak and buy a 3 or 7 day pass for the mass transit. You will more than make up the cost of these passes in the Euros you save with the cheaper room rate and the lower cafe prices.
There is no wi-fi but there is one free adequate-for-email computer with web access in the lobby.
In our case, we had a room above the back courtyard. Those that face the front of the building face the noisy trolley line, which goes by the hotel every 5-10 minutes. Ours did not, so it was quiet. (Other than the trolley, there is little car traffic on this street.) Our weather was cool, so I can't comment on how the room would be during summer heat.
One last piece of advice to US tourists -- Austrians are polite, hard-working people. They follow the rules and they expect you to do so as well. Don't put your feet up on seats on the bus or cross against the light. They will yell at you. And if you are in a hurry at at a restaurant or cafe, pay when food/drinks are served. Otherwise, you will wait forever for a bill.
This review is the subjective opinion of a TripAdvisor member and not of TripAdvisor LLC.