UN520,
You could also look at an alternative by staying within a chalet in a seaside area and use it as a base to either visit the mountain or other coastal cities.
For such locations, check:
Batroun
Jbeil
Jounieh
Beirut
Brummana is also another good location, while it is located at 700/800 m, it is still relatively close to Beirut.
This site will provide less expensive but still valuable accommodation:
HostelsLebanon.com
Hope that helps,
Cheers