OK, Israeli "bus expert" here!
When you say you arrive on Friday in Tel Aviv, do you mean Ben-Gurion Airport or the city? And what time on Friday? It makes a huge difference.
Public transport in Israel doesn't run on Shabbat - the Sabbath, meaning it stops running on Fridays an hour or so before sunset, and this time of year it that is about 4 p.m. That's the time the last bus reaches it's destination, not the time it sets out. Service doesn't resume until after dark on Saturday evening.
If you arrive in Tel Aviv (city) early enough in the day:
Bus 961 is from Jerusalem, not from Tel Aviv, and it would be going backwards to go via Jerusalem. Your best bet would be to go via Afula (major bus hub in the north of Israel). There are very frequent buses from Tel Aviv to Afula, a stopover on their way to another destination. They leave from the 7th floor in the Tel Aviv central bus station. The 7th floor of the bus station has two separate sides and you want the Egged side. If you end up on the wrong side you have to go back down to the 6th floor and then find your way up to the Egged 7th floor. From Afula there are buses to Beit She'an. You can see times at www.otobusim.co.il (click ENGLISH in top left-hand corner).
If you mean from the airport, this would be a bit more complicated. I'm at work now, but can look it up for you when I get home, around 10 p.m. Israel time (I work late on Wednesdays). I have heaps of public transport websites (mostly in Hebrew) saved on my computer at home.
If you arrive after about 1 p.m. on Friday it's a whole other story. You might need to take a string of sheruts (shared taxis), some of which run on Shabbat. Again, I have these saved at home. Probably - if you have to do this - the best way would be a taxi to Tel Aviv (no sherut from the airport to Tel Aviv), a sherut from there to Haifa, and MAYBE a sherut from there to Beit She'an. Not all sherut routes run on Shabbat. I have phone numbers saved (at home) and can give you these.
Please post again stating expected time of arrival in Tel Aviv, and whether you mean the airport or the city.