Internet Cafe's are available today in most of the places you'll visit, in the major cities its easier to find them as there are much more then in the small ones.
Here are two links that will help you to find a Local I-Cafe's in the main cities of Israel:
The Diaghilev Tel Aviv-Yafo
19 Reviews and 537 Opinions We were there for just a few hours in the night but we really enjoyed our stay. The concierge called...
Shalom Hotel Tel Aviv-Yafo
1 Review and 243 Opinions Shalom Hotel is located in Tel Aviv, facing the Mediterranean Sea and overlooking a garden and...
Center Hotel Tel Aviv-Yafo
1 Review and 172 Opinions The Center hotel is a nice little hotel in the Center of Tel Aviv, just by Dizingof Sq. Unlike the...