Odakyu Ito-kanko Freepass

We would like to visit Atami and Ito for a 2D1N stay and would be staying at Ito. However, we do not have the JR pass. Should we get the Odakyu Ito-kanko Freepass or is there a more cost effective alternative?

Odakyu Ito-Kanko Freepass looks like a set of tickets; a free bus pass inside the Ito area and tickets to get there. You seem to be just allowed to exchange trains at JR Atami station if you have to do that. So you should have to pay an extra if you want to walk around Atami as well.
http://www.odakyu.jp/english/deels/freepass/izu/

Tokai bus has their own passes.
1300 yen: Ito Free Bus Pass
700 yen: Atami Yu-Yu Pass
3900 yen: Izu Unlimited Ride Pass
https://www.tokaibus.jp/page.jsp?id=6241

Atami is covered also by Izu-Hakone Bus.
500 yen: Atami Mankitsu Pass
1200 yen: Zekkei Fujisan Pass (includes Jikkoku Path Ropeway)
https://www.ataminews.gr.jp/access/178/
https://www.ataminews.gr.jp/image.php?h=500&f=/db_img/cl_img/178/74ED9...
https://www.ataminews.gr.jp/access/179/

So you need to do the math depending on your own needs.

In the case of a sample departing from Shinjuku below, you could save 1100 yen (=1940+1300+2270-4400), with a cost of additional transfers at Odawara to ride on Odakyu trains.

D1 Shinjuku-Atami(Pass), Atami-Ito(+320)
D2 Inside Ito(Pass), Ito-Shinjuku(Pass)
4400+320

D1 Shinjuku-Atami(1940), Atami-Ito(320)
D2 Inside Ito(1300), Ito-Shinuku(2270)
1940+320+1300+2270

320 yen: JR Atami-Ito
1940 yen: JR Shinjuku-Atami
2270 yen: JR Shinjuku-Ito