Can upward-facing receptacles be installed flush to a kitchen island countertop?
The countertop issue has changed recently in the NEC code, so you need to find out which version is applicable for your jurisdiction (assuming you are subject to the NEC at all). In version 2011 and prior it was not permitted to install any face-up receptacles in an counter. Starting in 2014 (and the brand new 2017), you may install a receptacle in a counter only if it is specifically designed and listed for that application. (E.g. at random, this countertop pop-up receptacle).

The NEC does treat kitchen islands differently than regular countertop and peninsulas. You are required to have at least one receptacle. Normally outlets are required to be above the counter but for islands you are permitted to install the outlet below the counter (not more than 12" below), and there must be less than 6" overhang at that spot.

So if you are subject to NEC 2014 you could go with a pop-up receptacle. If you are on 2011 or older you must put one in the side somewhere. Under no circumstances are you allowed to install a regular outlet face-up.