I can safely tell you that yes, there is room for this in WoW.

Naturally there will be people who will shun you right off the bat, either because they think you're being impolite and haughty in an out of character way, and there are plenty of people who even if they catch on that you're RPing, probably won't deal with you too much because of the attitude.

If you're willing to persist, however, you can definitely make it worth your while. There are all sorts of characters out there from my experience, and even avoiding the dreaded cliches will probably net you a few people that will interact with you. I imagine you'd even find a number of people who might dislike you in character, but will likely whisper you once they leave and say "That was really good, I'd be happy to run into you and RP with you again!" A lot of RPers, more than anything, are happy to find some kind of conflict. If you can provide a character that might not just agree with them completely, they'll probably find you interesting and come back.

From a lore perspective, this is entirely feasible as a character, too. The easiest examples I can think of would be either Humans or Blood Elves. For humans, it's implied that there's a noble caste in Stormwind from the whole Defias chain, and you could probably draw from that to make your dysfunctional noble family. There are plenty of other human lands that you could hail from which might even be better for that: If you were, say, a deposed noble from Alterac (which betrayed the Alliance during the Second War), you could easily pass yourself off as being arrogant and haughty, annoyed at the fact that you've lost your lands and many of your rights despite your attitude, and even get annoyed that you're forced to work with low-life Elwynn residents. Plenty of possibilities there. Blood Elves are a little less established, but given that they were once the Highborne of the Night Elf race, it's hard to imagine them as being anything less than noble. Given that the Scourge sacked most of their lands only a few years ago in the timeline, you could easily have had your family shattered or shaken there. And even the other races probably have some openings. Warcraft has a lot of land and a lot of lore to play around with, and it wouldn't be too hard to adapt a piece of it to your character without breaking any rules.

If you're looking for a place to give it a shot, I personally recommend Moon Guard as an RP server, but that statement is tainted by favortism since that's where I play. But of all the servers I've tried, I've had the best response and found the best RP community to be there, particularly on the Alliance side. There are plenty of other RP servers, though, and you might want to try poking around a few and seeing if any interest you.

Don't give up hope. This kind of character can work, and I've even seen it work. It's not even that difficult to do, and can end up creating a really fun character to RP and RP with... even if, ICly, you hate each others guts.