Ditto to the you should be ok comments. A rough rule of thumb is that at 13,000 feet you start watching altitudes and how you feel. Above that height, it is suggested that you only ascend as little as 1,000 feet per day to allow your body time to acclimatize. That is hiking of course so on a bike the bad thing is that you can ascend way above that quickly but you can also descend before you are exposed to those altitudes for too long of a period. 180 miles is pretty short, you'll be through there pretty quick. Keep in mind that climb high, sleep low is something that all mountaineers follow. You expose your body to a higher altitude, that kicks the body into gear and you descend lower to sleep.
It depends on where you are coming from and the altitude you have been exposed to before you get there, but in general you should be fine. I would say that there is a very high probability that you will get altitude sickness (eg, headache, loss of appetite, insomnia) but not AMS (acute mountain sickness) of which the less ideal symptoms are pulmonary edema or cerebral edema (bleeding in the lungs or brain). You can die from those later things, but just be aware they exist. For most people, they won't be affected but I have seen a few people whose physiology simply didn't adapt to altitude very well and had to descend. That is the miracle cure if you are one of those rare people, you will feel it and you can just descend.
Have fun, I'm sure it'll be a beautiful ride.