Rats can get viruses but they are not the common cold like people get. It could be a Myco infection, or one of the many viruses such as Sendai, SDA, corona virus etc. Rats can go downhill pretty quickly so I would suggest taking the rat into the vet asap where they can prescribe a broad spectrum antibiotic such as Baytril or Doxycycline.
SDA virus does irritate the eyes as it causes the eyes and mucous membrane to lose it's lubricant which irritates the eye and can cause ulcers etc. They can give her an ointment called Terramycin which is used to keep the eye moist and avoid any furthur irritation.
How old is the rat? Has she had any new rats come into her home, or visited a pet shop and played with the rats at the petstore lately? Viruses can very easily be brought home on hands and clothing.
What litter is she on? Pine and Cedar can cause upper respiratory irritation. Aspen or Carefresh is a nice bedding to use for rats. Also why is the rat alone? Rats really do better in single sex pairs, perhaps after she is better they can look for a companion for the ratty.
Bookmarks