Can you put a bike rack on a Suzuki alto?

Yes.

google an example, say from UK - "Suzuki Alto Bike Rack" to see potential design outcomes.

Basically a tow bar suited to the Suzuki Alto is fitted (bar is suited to small box garden type trailers), on which you add a bike rack. Some designs are pretty nifty and include UNECE light functons; STOP/TAIL/FLASHER/REV/REAR FOG/RED REFLECTOR.

Cover the wheel reflectors (in Australia these are amber) where used domestically with tape when on a night journey. (Bike wheels on carriers can spin, and for folk approaching the rear at night, it can look like a bike crossing in front of them)!