To set the fork oil level:
Pump the damper rod up and down through it's full stroke until all the air is pushed out; the resistance will be smooth and even across the whole stroke when that is done.
Pump the fork tube up and down as well, to drive out any hidden air.
Compress the fork tube fully; there will be a bit of extra resistance in the last 10mm or so, which is what the oil lock pieces at the bottom do, basically a hydraulic bump stop so the forks don't bottom out with a clang and break something.
The damper rod should also be fully down.
Now set the oil height, standard is 62mm measured from the top of the fork leg.
After that drop the springs in, and you are correct the manual states tapered end up, which is opposite of all the other forks that I've looked at. No idea if Honda made a mistake or its meant to be that way for a special reason in the ST13. My logic has always been that tighter coils should go to the bottom as they get compressed first for suppleness, and the larger coils only come into play after the tight coils are coil-bound, so you get the smallest mass moving with the wheel. I can't see that it will cause any issue either way, as the spring is being compressed from both ends in a relative sense.