If I toss one whole hop

in a 12oz bottle of beer. how long before even the smallest bit of aroma starts to come around?

and how would you do this? swish it vigorously in a star san solution and then drop it in to the bottle?

I'm not sure if you're asking if you can dry hop whilst bottle conditioning or if you're going to open a beer and hop it. If it's the later, you're better off just sniffing a hop flower or pellet while drinking that beer. If it's the former, you're much better off hopping your beer in a secondary (or primary would work too) once the beer has fermented and is clearing out. You don't need to starsan hops since they are antibacterial by nature (they preserve beer). Just dump em in and forget it. Most people hop for 5 days or more. The aroma and oils will disperse in the liquid rather quickly, but to be integrated can take some time (so the aroma doesn't just fade away as soon as you open the bottle). It also will take time for the hops to settle out so you can rack into your bottles. There are plenty of threads you can search if that's your goal. Hope this helps.

I just heard of some company that tosses in a few whole hops in to their bottles. IPA of course

I wanted to try this on a pilsner and see if a slight bit more edgy hop aroma comes out after one week is all

I probably will still see what happens on one bottle, but I figured I'd query

It works, but the hop will get soggy and gross.

Soggy, gross and oxidated. Not a pleasant situation.