A 386 draws almost no power, much less than an Atom. At the end of the 386's market life, which I should point out ran for 22 years, it was being used in handheld devices that ran off 1 alkaline battery for months at a time, and those had radios in them.
Obviously you get more bang from something like an Atom but I don't think the 386 had any power saving tricks to learn from it.
Maybe this is related but by the end of its "computer life" Fujitsu was paying cash for any 386 you could get them (even used) because it was part of some network card they sold.
Obviously you get more bang from something like an Atom but I don't think the 386 had any power saving tricks to learn from it.