I toured in Israel unsupported in the previous century, so I don't know how useful my comments will be. I would urge you to look up weather data for the country. The climate is similar to the climate in California. Winter is the wet season, and the north is wetter than the south. It can get very hot in much of the country so I suggest that you avoid the warmest 4 or 5 months of the year.
The population of the country has grown a fair bit since I was there, and the tech industry has been very successful, so the country is more prosperous now. I suspect that there are a lot more cars on the road now than when I was there. Drivers were often not very good and there were few other cyclists on the road, despite a generally favorable climate.
The north was pretty nice for touring including around the Sea of Galilee and the interior, and I was able to find some pleasant back roads to ride up to Jerusalem from Tel Aviv. At the time I was there, the political situation was very different from now, and I biked in parts of the Palestinian territories. I also biked in the eastern Sinai, but I wouldn't advise that now. I now have a Bike Friday but I didn't have one yet when I was in Israel. I recall being able to take my traditional touring bike on the bus between Tel Aviv & Eilat. I had already been in parts of the Negev off the bike, so I didn't bike there. There are trains now which didn't exist when I was there.