Karla:
Suggestion... Change your title so that it matches your need...
Ex: How could Arduino monitor #hashtag on Instagram and automatically print picture?
Now, the bad news. You would need additional Arduino hardware, at a minimum an Ethernet shield.
You or a friend would also need to be able to code the parse routine to extract the hash tag target address. For example, here is code to parse a weather site:
http://forum.arduino.cc/index.php/topic,39023.0.html
Once you get the tag target(s), you must download the picture binary and manage the printer. Not a trivial task. A Due would have a better chance of doing the job, but if I were doing thus project, I would do it all with a Macbook or a Linux PC.
Ray
PS: have fun with your party.