First of all, you have to know which visa you have: work visa (B) or student visa (A). There is a big difference between them. Limor Levy, from the Visiting Faculty office (email: firstname.lastname@example.org, tel: 02-5882924) will help you dealing with the process of visa renewal. If you are holding an A2 visa, be sure to contact Limor at least 1 month before the expiration. If you are holding a B1 visa, be sure to contact Limor at least 3 months before the expiration. The secretaries in your institute shall take care of the paper work together with Limor.
Limor can only help out, but the whole responsibility to obtain, prolong, change your visa is on you! Be aware of your expiring dates!
From the Post-docs experience, it is very hard to make an appointment at the Ministry of Interior (“Misrad Hapnim”) online or by phone. You should go there in morning hours with your passport (ID) and schedule an appointment. Be careful of periods of holidays, as they can delay the process. Even though is unusual, visa appointments can be cancelled due uninformed strikes.
Make sure by yourself that you got everything you need on time for the appointment and ask again if you did not receive all the papers ~10 days before. The internal mail is so slow that papers cannot travel from Mount Scopus to Givat Ram in less than a week.
When you get your work\student visa, do not forget to ask for the multiple entrance visa (M) and fill up all the forms for it (multiple entries renewal requires an extra fee, so remember to take enough cash to your appointment). Otherwise, you will be able to go out of Israel, but not to come back, although you will still have your work\student visa. Note that multiple entrance visa (M) finishes a month before your work\student visa. Plan your trips according to these dates.
Be aware that you cannot switch from one type of visa to another (i.e., from work visa to student one, or opposite) without going outside of the country.