There is indeed such a place next to Ein Gedi, called Nahal David (David Creek). Though nothing is guaranteed with wild animals.
I am actually hoping to get a close look at Hyrax
A close look? How close? Your best bet is to look for a zoo that holds them in a small cage. Other than that, these are wild animals, and it won't be easy to catch a close look at one. I heard that there are some in the Ramat Gan Safari, though I could not confirm that from that website - you want might to call or email and ask.
That said, the Rock Hyrax is a common animal in Israel, and I've seen families many times, especially in the north of the country. If you go hiking in the Galilee or Carmel region you are likely to find some. However, if I was actively looking for them, I'd go to a colony whose location is well known. Examples I know of:
- Nahal David (David Creek) - that's right next to Ein Gedi. I can really recommend this - I've personally seen Hyraxes and Ibexes there, pretty close by. So your choice - Ein Gedi - is actually great in that regard!
- Rosh Hanikra - there's a colony on the rocks near the parking lot, and more on actual rock formations above. You can also see them from above when you approach the entrance to the site (after the parking lot).
- Kibbutz Shamir - there's been a colony there for ages, from what I understand.
- Rock Hyraxes are mostly active in the mornings, I've read that sunrise to 9 am are the best times to spot them.
- The Ein Gedi area is absolutely breathtaking, but it's not a particularly good hub for touring the rest of the country - Tel Aviv or Jerusalem are better for that. Though admittedly it's overall a tiny country.