The only way to get to Dan during Rosh Hashana is using a taxi service. It will take you a full day because you'll need to take a taxi from Tel Aviv to Haifa then look for a taxi from Haifa to Qiryat Shemona. Ask at the information desk at the airport. This service is not very expensive. I'd guess it's ~NIS 60/person from Tel Aviv to Haifa. The same price maybe more from Haifa to Qiryat Shemona.

Bus # 842 from Tel Aviv to Qiryat Shemona runs on Saturday from approx. 15:30. In Qiryat Shemona take a bus or taxi to Dan, and start on Sunday morning in Dan.  The gate from Dan to the INT is closed until 08:00. 

An alternative is to start hiking in the area of Tel Aviv. On Friday and Saturday complete day 20s, and 21s in the guide, both are easy days. On Sunday take a bus to Qiryat Shemona and continue from kibbutz Dan. When you get to Tel Aviv skip the two days you've hiked and continue south.

Enjoy the hike on the INT.