Yes, it is possible and JST have a hand tool specifically for these terminals, it's part number YRS-859. It is not cheap, but is the only way to get repeatable, reliable crimps, especially on terminals that small.
You can use generic crimp tools Engineer PA series sold by Adafruit and others, but at best those crimp tools make for a very fiddly job on terminals as small as the SH, and in many cases they either deform the terminal or don't compress it to the right dimensions, so you have to do a lot of fiddling and squeezing the terminal in different ways to get the terminal to fit in the wire housing. I've tried those crimpers on 1.5mm terminals for which no proper hand tool was available, and while it was doable, it was not fast or fun. For larger terminals that have wider tolerances they can work okay, but trying to get good results with 1mm terminals is going to be a tremendous challenge.
The proper, purpose-designed crimp tools have a number of advantages:
- the dies are designed specifically for the terminal in question and will reliably form the finished terminal to the correct shape
- Both the contact flaps and the mechanical flaps will be formed in one operation
- the mechanism will have compound leverage and a ratchet mechanism to assure that the correct crimping pressure is applied
- there will be locating features to hold the terminal in the proper position for crimping (and often to hold the wire at the correct depth as well), making it much easier to apply the terminal without four hands
So in short, if you only have one or two terminals to do, and it's to repair a personal item or something, then by all means, try the generic crimper and just be prepared for it to take a bunch of tries to get it right. But if this is for commercial purposes, or for anything where the reliability of the termination really matters, the official tooling is really worth every penny.