Gap between sash frame and pane of single-hung window sash - DIY-able?
Is it DIYable? Yes. No. Maybe?

You may find that you start the job, everything works out just right, and you complete the work in record time and under budget.  More likely, you'll run into a bunch of unforeseen problems, take 5 times as long as you'd expect, and end up way over budget.

The best way to tackle DIY work that you are not sure about, is to have the resources available to quickly fix mistakes.  In your situation, it might be a good idea to purchase a replacement window before you begin. If you damage the old window beyond repair, you'll simply have to install the new one. On the other hand, if the repair works out, you can return the new window you purchased. 

It also never hurts to have somebody with knowledge in the type of work you're doing, so if you run into trouble they can offer guidance.

If you come across something you are really not comfortable with, there are always professionals available to do the work for you. You might also be surprised that they charge less than you might think (not always the case, but does happen).