Hi Ashwinir18 -
Welcome to my Swiss Alps.
That looks like a good itinerary, as it stands. One thing might be changed. If you wish to go up to Schilthorn, you will save a lot of time by staying in Lauterbrunnen or Wengen.
This map will help make that clear:
Zoom in to see the transport tracks and stops.
This will help with timing and schedules:
There is a cableway connection from Grindelwald up to Männlichen and down to Wengen, which could be a nice way to get to Wengen from Grindelwald. It starts on the 19th. You are in the area at the tail end of low season, but several weeks before the Summer season begins. Not all restaurants or hotels will be open yet.
Another way to get more flexibility is to skip the Rhine Falls. Nice falls, good flow in the Spring, but Switzerland is not the only place in the world that has waterfalls.
Regarding weather... all high Alpine areas depend on good visibility for maximum enjoyment. In May, there is a significant chance of cloudy weather on high. Don't bother to go up to Jungfraujoch, at considerable expense, if all you will see is the inside of clouds. The inside of clouds is boring, in my opinion.
Don't buy a ticket for the Jungfrau untill the night before. In May, it will not be crowded.
Check with tourist offices, train station staff, etc.
Here is a link for weather -
If you have to be at the airport by 1:25 PM, you can do it, but you'd need a very early start. You could leave your luggage at the airport. If you are on Swiss International, you can check luggage at trains stations... not sure the advance time needed.