This is a very tricky question - in May, in the mountains, anything can happen weather-wise. Generally speaking, you'll be looking at highs in the 50's or maybe low 60's. It will probably be be below freezing at night, at altitude. You are a little early in the season. There will definitely be snow on the ground as you go over the pass, but the roads will be clear.
It sounds like you'll be coming across US 50, perhaps from Gunnison to Salida - can you provide some details about your route, and we'll do our best to give better guidance.
We all ride our bikes in the mountains in May. We hope for the best, and bring lots of layers of clothing.
At least, in my case, I do after I learn: my worst cycling experience ever was a return down Berthoud Pass. I rode from Idaho Springs to the top, and when I turned around, the weather decided to greet me with a windy rain-snow mix. I have never been so cold, nor so happy to finally make it down to the small coffee shop in Empire, where I huddled next to the stove for an hour, trying to see if my extremities were still attached. After that experience, I always bring appropriate rain gear, a hat, something to cover my feet (baggies work), and maybe a fleece vest. That day, I did not.