how to cool/heat this room?

Find out from the super how others do it -- presumably this is not the only apartment with such a window. You might get permission to modify the window to remove a couple of panes and vent a freestanding AC unit through the resulting holes. (Looks like R2D2 with a tube or two out the window.)

There are also ways to mount vertically-oriented AC units in old casement windows -- I know that's how it generally works at Hudson View Gardens (Pinehurst Ave between 183 and 185), for example.

How did it end up without heat? That seems unusual for a bedroom -- was it partitioned off from a larger room?

this was a combination of 2 apartments, so i don't know if there are other units like this. the owners lived here for 10 years, and they used the room as a study and more recently as a baby's room (i don't think during the summer), and they found that the air circulation from the rest of the apartment was enough to cool/heat it.

You can buy a portable unit to do the job especially if it's a small room because the cooling power is not very good in general. The upside is that they're easy to roll away and store in the winter time.

i don't know how you could do a portable unit because it is a casement window-- i.e., one big pane of glass that opens side to side like a door. is there a way?

If it is as you say a small casement window, there are casement air conditioners made for that purpose. Just type it into google. They come with panels that fill in the remainder of the window. Those tend to be white plastic -- you may want to get someone to install it, or if you are remotely handly get a piece of clear plexiglass to fit in the opening so you still get light. If this is a divided light casement window, you can get a portable unit and just take out one of the panes to exhaust.