I have been through this exact situation before. In my case, the company did well, had an IPO, and currently trades at more than 10x my strike price. I hope the same for your company, but you should not kid yourself about the possibility that the opposite will happen. This stuff is risky, but with great risk can come great reward.
Assuming the stock's fair market value (FMV) will only go up between now and the IPO, the best time to buy the shares is right now. The difference between your strike price and FMV counts as income for the AMT, so buy when the FMV is low to minimize your taxes in the short term. In the long term, you can actually get this AMT refunded to the extent that your AMT in future years is less than your regular tax. If you have never owed AMT before, you may be able to lower your tax burden by spreading your exercise events over two tax years, because you'll have some "wiggle room" where you could buy a few shares without owing any AMT at all, so being able to exercise a few shares AMT-free is a good idea if you can manage it. Balance this against the likelihood that the FMV will increase before January.