I'm aware of one in vitro study (though there might be others) of that demonstrate that caffeine is effective at certain types of hair loss -- in a "test tube", anyway. That is, this study from 2007 (another link from publisher, a colloquial summary of the study) took biopsy samples of follicles from men with AGA ("male pattern baldness") and treated them caffeine (and hormones), finding that caffeine was "a promising candidate for hair growth stimulation" and "may have important clinical impact". Another note is that this is ex vivo -- which I take to mean topical in this sense -- drinking a bunch of coffee is unlikely to help (the summary article suggests that the dose necessary of orally-consumed coffee to get the same effect is ~6 grams -- many dozens of cups of coffee, and perhaps fatal dose). If you don't happen to subscribe to the International Journal of Dermatology you can obtain the PDF of this article for perhaps a small fee.
I don't know if this has been demonstrated in any clinical trials. I can't find any longitudinal studies of longer-term effectiveness, nor can I find any evidence of actual clinical trials. There are a few more links in the caffeine-related clauses at the management of hair loss Wikipedia article.
Also, as an aside, you might get a little "caffeine buzz" from it! There are some bar soaps, like this one, that make the (largely unverified) claim that some of the caffeine is absorbed through your skin. Win-win?
"To keep this on-topic..." As for using instant coffee or some other source of caffeine (e.g., pills) to be applied topically to your scalp, I can't find anything about that either. Concentrations cited in the study are "0.001% and 0.005%." Coffee beans themselves have about 1-4% caffeine before roasting, so it's reasonable to expect that you could get to the concentration used in the study with a paste of instant coffee, but I don't know about the "bioavailability" of caffeine from topical application of instant coffee. Robusta beans are generally on the higher end of caffeine content and might be cheaper. Also, green beans will have a higher caffeine content than roasted, but I might be taking this a bit too far...
Standard disclaimer -- this isn't medical advice, I'm not a doctor, seek actual legitimate advice if you have health concerns.