You can easily take a taxi to XiaoXiMen 小西门 (West Inn), and take the Number 1 shuttle bus for 25rmb. It starts at 5am, leaves every 30 mins, and takes about 40 minutes to get there. Last time I used it (a few weeks ago) it left right on the hour, 6am, and I was at the airport by 6:40am.
Same deal on the return trip. The bus runs until 2am I think.
Number 6 bus also starts at 5am from the Kunming Hotel, which might be (slightly) closer for you, although I haven't taken this one, so can't vouch for it.