I wear monthly disposable contacts (Proclear is the brand and is claimable under my health fund), they are made for air conditioned environment. I also carry fake tears or rewetting eyedrops with me, in case my eyes feel dry. Your eyes will get a bit redder up in the air due to the pressurisation, but that's normal, even for people who don't wear contacts, they get red eyes. As long as when you get down on the ground, your eyes get back to normal whiteness, then there is nothing to worry about.
Wear what you wear now to give it a go, if your eyes get really red and don't return to normal, then you need to consult your optometrist, they will give suggestions on what to wear. Solution is also important, make sure you use moiturising ones, I use optifree, quite good. Good luck and happy flying.