I grew up in Vienna and visit family and friends every year, so while not as knowledgeable as a true local any more, I still know my way around...
OK, so you definitely have to go to Esbjerg in Krugerstr. in the first district - exact address:
Krugerstrasse 7
1010 Vienna | Austria
It's really close to the Oper and to Stephansdom, so you walk almost passed it anyway during your sightseeing and shopping - I would almost bet, that SWMBO likes it there as well - it's a fantastic store with everything shaving and scent related and the owners are super friendly. But if worse comes to worse, a lot of the major shopping is situated in that area as well, so SWMBO will definitely be happy...
The Haslinger factory on the other hand is not in a part of town you would normally go to (10th district) - but if you call/email ahead and order before you go, you get 5% discount of the retail price. So if you want to do a bulk purchase, it might be worth it. If you want the Email address of an English speaking person there, please PM me and I can help you.
Else Douglas Parfumerie in the City (they have a couple of locations close to Esbjerg and on Mariahilferstrasse - see bellow) carries most of the Haslinger series for a fair price.
Muller Drogerie (also multiple locations, Mariahilferstrasse is a big one) is your best bet for "cheap" supplies - we are talking Speck, Tabac, Palmolive, Weleda, etc
Esbjerg has the best brush selection as well - but at a price - and they do have other hardware also, but if you are after Merkur/Muhle razors and Solingen straights, as well as great manicure/pedicure items and knifes in general - I can highly recommend:
LORENZI Int. Stahlwaren - Siebensterngasse 41, 1070 Wien, Austria
It's a tiny store with LOTS of stuff - just off Mariahilferstrasse (the other major Shopping destination in the city) and carries everything knife related and they hone knifes as well (not sure about straights).
That's it of the top of my head - if you need any more info - just shoot me a PM