CTU International (1) and Domestic (2) are separate terminals. I presume you're talking about a domestic CA flight; there is a dedicated Air China lounge. I've never tried but presumably you could enter upon arrival. The Terminal 2 Air China lounge is probably the best lounge in the airport. They serve food during specific hours only. Should be serving breakfast for your departure.

Incidentally, CTU International Terminal 1 has a pretty poor quality Air China lounge. UA uses the slightly better Priority Pass accessible lounge instead.

Chengdu is in the process of replacing the airport with a massive mega airport connected by express train to Chengdu Nan Station. So I'm not sure how much more infrastructure will be going into Shuangliu, but they did build out the Metro to the airport which opened earlier this year. I usually just take Didi.