My mom goes, "The size of a city block?! And we almost didn't SEE IT???". Yeah, that's microscopically small to astronomers, and it was never closer than 100 million kilometers. And hey, remember that passive listening device scheme re: U.S. diplomats in Cuba and that seal thing we got from Russia (which still still cracks me up, btw)? What if this was the astronomical equivalent, and instead of a microwave source, it did something clever with the blackbody radiation spectra that we know all stars produce? Maybe it'd have to target only main sequence stars or something. Why not tumble the whole time? It'd be a clever way to ensure that every point in 'Oumuamua's visual field faces some preferred plane of R̶a̶m̶a̶ 'Oumuamua (you are getting very sleepy). But 8 hours on astronomical timescales gets the same reaction from astronomers as the football stadium size. Tumbling refers to the type of motion, though, not the rate. Maybe it would have to stop tumbling to effectively phone home though, that's an interesting thought. Sucks 'cuz the plane in which 'Oumuamua's hyperbolic path lies is far enough from the ecliptic plane (~20 degrees) that gravity assists aren't so much on the table. So it's way harder to find a trajectory that gets us there quickly, if one exists at all. Falcon Heavy with the lightest spacecraft possible, please. So we're already in the several hundred million$ price range. Woof :(1) From an astronomical standpoint, it ain't a big thing. We're talkin' about an object around the rough size of a football stadium.
2) If I had something shaped like a giant cigar and I were trying to not get noticed, I'll bet my AI would go "quick, act like an asteroid" as soon as I had a reasonable read on my observed race's intelligence. "Tumbling" is such a nasty word. It's got a periodicity of 8 hours. Sucker's "tumbling" about 1/10th as fast as your average revolving restaurant.
3) We should totally send something after it. That's the sort of speedboat space exploration your average Red Stater can get behind.