Well, I'm sure this is a stupid question but just in case my preconceptions are all wrong – would someone with a very obviously Jewish name encounter lots of difficulties working in Dubai? Or even be able to get a job/work visa at all?

There are plenty of Jews doing business in Dubai. One example, the 1539 room Atlantis is a joint venture between the Government of Dubai and the Jewish South African hotelier Sir Sol Kerzner.
As this is a travel forum, i would suggest you post question at Expatforum.com/dubai. You will get lots of answers there from residents of Dubai.

It makes absolutely no difference. Dubai is concerned with one thing -- money -- and anti-Semitism isn't profitable. Most of the people responsible for hiring are other Westerners, in any case. Also, there's no reason to assume that most Emiratis will have any idea that your surname is "obviously Jewish".

My husband has an obviously Jewish name and only person (Turkish) ever seemed to notice that it was a Jewish name. Because his handle (which is a portion of his name) - he was invited to join a minyon there - but of course the invitation came from other Jews.
I was somewhat concerned that our marriage certificate (we were resident not tourists) showed we were married by a rabbi; but if anyone noticed they never mentioned it.
Thanks for al the input. I was also wondering about the likelihood of encountering any prejudice on a daily basis???
I will lok at that expat site as well.
UAE does not acknowledge Judaism as a religion and it also does not recognise Israel. The only problem I can think of with getting residency and work permit is if legal documents or educational certificates came from Israel as they would not be accepted. An Israeli passport is a big no no. Having a Jewish name should be no problem.

You will hear anti-Jewish comments (they are really more anti-Israel but the lines get blurred). The majority of people understand the difference, but some do not. You will need to keep quiet about these and not comment (at least most of time). But prejudice against you? No, not at all. Mostly you just keep quiet. (We do think that my husband became a defacto Catholic - I had put that, truthfully, as my religion and then we never were asked for his. I am guessing it is listed as Catholic.
Wandabug - are you sure UAE doesn't recognize Judaism as a religion? I really don't think that is correct - although there are no formal synagogues in the country.