There's a no-immigration transit if you're checked through. You'll have to go through a WTMD and have your carry-ons X-Rayed again. It's rather fast, though. They do run the machine readable portion of your passport through a data collection terminal attached to the WTMD, but I never really seen them inspect it other than to compare your picture, so have it and your BP out ready to show. If you're in the front of the plane, and it really is early, I've gone through it in about 30 seconds flat, mainly to discombobulate and recombobulate. (edit: by that, I mean, "pockets and outerwear")

If you think of Terminal 2 as a big letter "H", transfer security is at the cross-bar of the "H" on the Arrivals level/2nd floor. You'll pop out of security in the cross-bar of the H on the 3rd floor. Go up another floor for the EVA lounge. The Infinity lounge for J passengers is on the 4th floor, in the "H" cross-bar, "in the back". Against the wall, not the one in the center.

1:30 is a bit short for a shower, depending on the time of day. Early mornings, it'll be enough. Daytime/late afternoon, you'll need 2-3 hours for the queue to filter through. Early mornings, :30 to 1 hour to get in. (my flights from the US usually show up right at opening of the lounge, or soon after.

You'll get a pager in exchange for your boarding pass. When you're done with the shower, you'll turn in the shower key and get your boarding pass back. If you have to leave before your turn comes around, exchange the pager for your BP.