Weeping cherry grafting

Its certainly possible to prune it as long as you have some sharp secateurs - I'd leave it for another month before cutting anything though.

Probably being a bit thick, but I'm not sure what type of grafting you have on mind and what the point of it would be.

I saw the same tree top grafted and it looked like the one below. Basically, it is a weeping tree and the branches are getting longer and lower ones are touching the ground now. I want a fuller look with all branches stemming from the top
To be clear, I want to cut the branches and graft them to the top.

OK I'm with you now - you want a sort of umbrella effect with a clear trunk and a number of branches then at roughly the same height. The good news is you don't need to do any grafting - the downside is it will take several years to get to where you want to be with it.

So in about 4 weeks time take off all the side branches cutting parallel with the trunk (small saw ?) up to the height you want ( tidy all cuts with a v sharp knife). Cut off the trunk above the branches that are left. Depending on the height you choose this should leave you with the trunk and 2 or 3 branches only. What should happen then is the tree responds by staring new growth both from the branches and from the trunk - if this is where you want it to be OK, if not rub out these shoots using your finger and thumb while they are soft. In time some of these new shoots will become branches to gradually develop in to the shape you want.

Please note that the initial cutting back operation does involve a small risk of killing the tree - by leaving it to June you will have minimized this chance, but it is still there.