Here are your choices -
* If you fly non-stop from the US your flights get into Mumbai between 9 and 11 pm. (DL, AI, CO). You do not have an easy air connection to Pune until next morning. So you could overnight in Mumbai and take the first flight out in the morning. Or as most people do, hire a private taxi that takes you from the Airport to Pune in about 3 hours. Most of the trip is on a expressway comparable to US highways. The same trip will take 5 hours by road during daytime.

* The privatair flight is a Lufthansa all business class flight from Frankfurt and lands in PNQ. But it does not have good connections to/from FRA to the US. If you can overnight in FRA or find a reasonable connection, this would be one of your only options to go directly to Pune.

* Most Atlantic routes with 1 stop in Europe also get into Mumbai between 9pm-1am. So no air connections. But there are a few options with BA, Virgin, EK, Qatar which land early/mid morning and do have air connections to Pune.

I have done all the options multiple times and can further help you if needed.