Vienna Hotel

There is the Vienna Hilton next to the Wien Mitte train station. The Pension Stadtpark is a block away for less money.
Hint: Use the S-Bahn from the airport to Wien Mitte. It is a lot cheaper then the express City Airport Train. Unless you are in a hurry, the City Airport Train takes 16 minutes into the city, while the S-Bahn takes 25 minutes.

Looks like the first S-bahn is at 4:19 getting you to the airport at 4:45. Otherwise might want to stay by the airport.

If you stay at the airport ( which would be my choice ) , there is a very nice NH hotel a very short walk from the terminal

At the Vienna International Airport you have the choice between the NH and the MOXY, both in walking distance of the terminals.

If you do not want to stay at the airport, the train ride with a RailJet from there to Wien Hbf (main train station) is 16 minutes only, no stops inbetween.

Around the train station you have several hotels, two new ones just opposite the south exit.
http://hauptbahnhofcity.wien/hotels/

From main train station you have several public transport connections, e.g. metro U1 to St. Stephen's Cathedral (historic center) in 5 minutes.

There is no danger in the area around the train station.

In my humble opinion, you cheat yourself if you stay at the airport, especially if this is your first time in Vienna. My suggestion is to find a relatively inexpensive room in/near the middle of the city and then spring for a cab to the airport the next morning. That way, you can wander the streets of Vienna as late as you want or even catch a show at the Staatsoper or a concert at the Musikverein (sleep on the plane!). I have done this before by staying at the Motel One Wien Staatsoper, which has a good location, one block away from the opera house outside the Ringstraße and right around the corner from the Karlsplatz U station. It also has A/C which might be needed in July. Usually, you can get a room for 70-80 €. I don't remember the cost of my cab, but I don't remember it being ridiculously high.

and then spring for a cab to the airport the next morning.

The problem with a cab is not the price, which should be € 30 flat fare, but the morning traffic.

There are three different train lines going to the airport. The rides are between 16 and 25 minutes depending where you board.

Traffic for a flight that departs at 6 am? Nah.

Or stay up all night as in "Before Sunrise" ...

Traffic for a flight that departs at 6 am? Nah.

Sorry, I overlooked the departure time.
By the way, at that early hour only the S7 trains goes to the airport, not the RailJets.