Would not prefer to spend the money...
Well, ...you should consider the total cost of leaving the country to apply for a new visa, against simply changing your flight. Throw yourself on the mercy of the airline - explain that you need to leave India because your visa will expire.
My multiple entry visa expires on the 4th June and if I apply for the 90 days visa now, I would have to leave India by 1st July. The problem is that I don´t really know how I can get this 90 days visa.
I take this to mean that your visa (whether it was for 12 months or 6 months) expires on the 4th of June. You cannot apply for a new visa to India whilst you are still in India.
The problem is that I don´t really know how I can get this 90 days visa
If your visa is valid, then your next 90 days stay would commence when you re-entered India. But you are saying that your visa will expire (not just the 90 days entry).
Would this be easier in your first language?
https://forum.lonelyplanet.de/viewforum.php?f=66
I am still unclear about your problem. Go to Nepal and apply for a new Indian visa whilst you are there. There is no guarantee that you will be issued a visa of sufficient duration. Either change your flight, or keep going back to Nepal.
In Nepal you need to apply for the visa here:
http://nepalsbi.com.np/content/indian-visa-service-center-ivsc.cfm
If you need a 60 day double-entry visa with which you must arrive by flight, then consider:
https://indianvisaonline.gov.in/evisa/tvoa.html