When a ganglion cyst burst, is it possible to feel sick afterwards?

Anything is possible: But unlikely. Ganglions are filled with mucin, which is similar to synovial fluid found in joints. When they "pop" the fluid goes into the surrounding soft tissue and subcutaneous space. Very little if any of the fluid is immediately absorbed into the blood stream. Instead, the fluid is resorbed slowly over time. Feeling sick at the time is likely coincidental, or a vagal response.