Can I take a Schengen Visa through the Consulate of Italy (for the purpose of my MS) and then visit Cyprus for a few weeks and then continue my studies in India?
The terminology is sometimes a bit confusing but the visa you will need for your studies in Italy isn't exactly a Schengen visa, it's a national visa. It also gives you the right to visit other countries in the Schengen area (and a few candidate countries) for up to 90 days. A visit to Cyprus after starting your studies may be possible but not a long(ish) internship.
Furthermore, even if a multiple-entry Schengen visa does exempt you from the requirement to get a visa for Cyprus, this visit to Cyprus must be incidental, it's not a valid reason to get a Schengen visa. The Italian consulate only issues visas when Italy is the main destination and you cannot expect it to grant you a visa just because you want to go to Cyprus.
The visa you will get from Italy will therefore be based on the documents you submit regarding your master's studies in Italy and is unlikely to start several months before the course begins. A visit before the beginning of the academic year on that basis will be impossible.
If I can, then can I stay for more than 90 days in Cyprus?
No, it is not allowed to stay longer than 90 days in Cyprus on an Italian visa (of any kind). You can however visit Cyprus for up to 90 days with a long-stay visa or resident permit (which you would get for your studies). Do check whether you need some sort of work permit for the internship you have in mind. Even if you are allowed to visit Cyprus, you may not be allowed to complete a (paid) internship without additional permission.
If you would be in my place what approach would you take to get to visit both of these countries?
It's a little tricky. The by-the-book approach is to apply for a Cyprus internship or work visa and as soon as you move there, apply for an Italian student visa from the local consulate. The problem with that is it will be difficult to get additional documents if needed, doesn't give you a lot of time and can get very costly if you don't get your visa.
I assume you have already secured your internship but if that's not the case, it might make sense to do the Cyprus internship at another time. Even if that's not possible, it might still be better to just give up on the internship to make sure you are able to join your master's programme in September.
Alternatively, if you have to go to Cyprus for less than 90 days, you could apply for a Cyprus short-stay visa (it's not a Schengen visa but it's similar). The problem you will face is that, unless you are able to get your Italian student visa well in advance, you may need to go back to India to submit your application and collect it from the Italian consulate there.
