Swapping in Type R engine isn't the same as swapping in an old LS motor. The B18a is a OBD0 motor that comes with a cable tranny.
But yes you can swap a b18a into your Civic. You'll need to convert it to OBD1 and either get a hydraulic tranny or get the conversion from hasport. If you can't find any info on this swap, search for the b16a SiR ('88-'91) swap and just skip thru the vtec wiring. Also check with your state on whether it's legal for you to swap in an older engine into you car. Some states don't care, but others do (like Cali).