Nope, sorry. The only ways of getting to the airport that I know of are:
- Train (DOESN'T RUN ON ROSH HASHANA)
- Bus (Egged isn't even giving an option of travel on Thursday. Interesting, because they ARE giving Friday and Saturday as options.)
- Sherut (Only Nesher to Jerusalem and Amal to Haifa)
Your best options are
rental car and
taxi.
There are a bunch of taxi companies in Haifa that do operate to TLV, but I'm not sure which are better priced or more reliable. Are you staying at hotel, and are you in Israel yet?