Watching The Ashes in Beirut

You could check the big hotels.

And there is an Irish pub (Celtic?) you could try.

I'd say your best bet would be to find somewhere with a high concentration of Lebanese-Australians - I can't imagine anyone else would be particularly interested in the Ashes. I don't know if there's anywhere in Beirut which is particularly known for having a lot of Australian-Lebanese, but every third person in the Qadisha seems to have links to Australia - I'd say you wouldn't have to search too hard to find somewhere showing it in Bcharre, Ehden or Kfar Sghab.