JST 1.25mm connector soldering
Is this connector meant to be soldered the way I described


Not really, but don't let that stop you.


  or is it just a through hole connector meant to be soldered with a vertical orientation? 


Yes.


  If the former is true, how to solder it, just put enough solder paste/tin under the pins to fill the space?



Put pads on the edge of your board, something like the edge of an ISA PC card.
Put one or two plated through vias in each pad
Hold connector in place, flush with board
Solder


Solder will wick into the vias, and this will increase mechanical strength. It should be a bit stronger than simply relying on pad adhesion onto the PCB.

This isn't something you'd do in a fabrication run, because it has to be hand soldered, but it'll be much stronger than a wimpy SMD connector. Like the micro-USB in your smartphone/tablet which falls off the board if you plug in the charger wrong.

I do it all the time with 0.1" headers for prototyping, and the headers will bend before the solder breaks.