To be clear, there is no refrigeration of the sort you require and while there may be a FA willing to place your medicine in an iced area, that would be in violation of UA's food safety rules for fairly obvious reasons.
You should consult the package insert or label (meaning everything that accompanies the drug) for any information it may offer and, if nothing describes what you require, speak with your physician, pharmacist, or even the manufacturer.
As others note, many medications requiring temperature control are shipped routinely in styrofoam containers with a cooler pack (not an ice pack because that may cause freezing) and are in transit for 24-48 hours. Perhaps your drug can tolerate that? This presumes that the container is not opened, meaning that what you need for the flight itself will need to be more accessible,
It is particularly important to pay attention to minimum temperatures too. Your drug may tolerate sub-freezing temperatures such as might surround a cart on a ULH flight, but many will not.
This should all be fairly easy information to obtain as tens of thousands of passengers require some temperature-controlled drug and manage.