The process in Israel is commonly informal and less structured than e.g. the US process. The nominal scenario is: you contact a potential advisor(s); if they are interested they will tell you to formally apply and tell you if they can afford funding; your application will eventually get to the university's admittance committee which in turn will just reach out to your potential advisor and follow their recommendations.
The "formal" process can take as little as 2 weeks and as long as a couple of months. Further, there is no obligation to start Phd studies at the beginning of the academic year, so I've heard of (non-foreign) students who applied two weeks before the year starts and were able to start on time. I also know about students who applied a couple of weeks after the beginning of the semester, who were admitted as well (maybe their "clock" started ticking at the beginning of that year, so they lost a bit of time).
For international students, you need to allow 2-3 months to obtain student-visa and arrangements, so the schedule should be something like this:
- Jan-Mar: initial contact with advisor, check if there's an opening
- Mar-May: making sure the advisor wishes you to join their group and has secured funding for you. Good time to start meetings online with the advisor to see that you get along (and for them to see that you are a good fit for them).
- June - Formal application
- August - Formal university decision
- September - VISA, flight tickets, lodging arrangements
- October - arrival / school begins
Again, this is very fluid, and you can skip to contacting the advisor at May (but maybe the position will be filled by then and no more funding will be available for you). If the advisor wants you, they can usually make the university admit you out-of-schedule. Do check in your desired university's webpage for international students' information, since some universities might have different rules and schedules.
Best of luck!