With any swap short of the H-series and K-series you can keep your A/C. That means that with any b or d series motor you can stay nice and cool.
I will argue that the b16a SiR is the best motor for the 2nd gen CRX. That is because the motor comes stock in Japan and therefore is well balanced for the car. Sure its a torqueless wonder, but any large amount of torque in a CRX just means you'll be burning through tires. However, the main disadvantage of the b16 swap is the price. To get everything installed and finished I payed about $3000 for my swap. I would say the cheapest you could do it would be $2500. If you go this rout you'll need aftermarket mounts, and I STRONGLY suggest hasport (although Az Racing has some nice looking billet alluminum ripoffs of the hasports).
The second choice would be the b18b from the LS integra. No VTEC, but it does produce more torque and a pretty decent amount of HP. If you want to boost, this is the engine to go with. You should be albe to find one for arround $500 in a junk yard. You'll want to avoid the LS transmission however, because the gearing is really shittily long. I'd say, get a b16a tranny from a SiR and use that mated with the b18b for max potential. Since this is also a b-series swap you'll need mounts just like above.
The third choice is the ZC. Best one to get is the DOHC non VTEC, but it also comes in SOHC non VTEC and SOHC VTEC. The DOHC is the best motor though and provides the most tuning opportunities (independant intake and exhaust cams) if you want to go turbo. The best part about these engines is how cheap they are. You should be able to pick up a ZC engine and tranny for less than $1000. This swap doesn't require mounts and will drop right into your car.
Now, there will be some differences in how easy the swaps will be depending on whether or not you have a Dx, Si, or HF. Dx you'll need to convert from DPFI to MPFI, and with the HF you'll need new front hubs to fit the larger axles. For all the swaps you should be able to use 90-91 Integra LS axles and halfshaft.