Sounds like the east side Big Meadow approach, given the distance you mentioned. The standard gooseneck route is a lot longer hike in (about 23 miles or so), but that climb itself is a lot easier. The usual summit climb from the east side is more technical , and there are various other hard rock routes that could be done.
If he's doing the standard Gooseneck route, no reason why you can't go to the top with him, depending on a few things (your level of fitness, your equipment & skills, the weather, current snowfield conditions, etc.). If the snow conditions are good (earlier in the year is better), the summit is a pretty non-technical snow climb, with a bit of rock - but nothing like you see in Jeep or Subaru commercials
hTo go to the top: in addition to being in good shape (see below), you'd need to pick up some basic mountaineering skills. If you've got a couple extra days in WY, look up these guys: http://www.exumguides.com/summer.shtml. Their snow school would probably be most applicable.
Of course, that fear of heights might be a real show stopper as far as the summit goes, but you can be your own judge of that.
The hike in will be a slog , but the scenery is good and your group should be able to set a decent pace. As noted in other posts, bugs can be somewhat of a problem from July until September. Not real widespread, but you get hatches where they're thick for a few days, and it's definitely worse around water. Wind is your friend when it comes to mountain bugs.
To get in shape, you should hike uphill (if you have hills nearby
hour ). On the other hand, everybody in the group will be sweaty and winded just like you are - if not, they're not carrying their fair share!!!!
Other useful gear.... bring a pair of gaiters, especially earlier in the year - essential for snow - they also help keep seeds and pebbles outta your boots. Earplugs in case the wind blows or your tentmates snore. Foot powder or talcum powder - good for the feet as well as other sweaty body parts (nutsack). Paperback book in case you're stuck in the tent. Extra camera batteries and memory card for digital camera, if you have one. Small package of baby wipes. Sunscreen, lip balm, and all the stuff that other people will tell you to bring. It could be 80 degrees and sunny one day.... a blizzard the next.
Fishing - you'd probably get a few chances to make some casts at the lower camp - upper camps are generally right where a glacier ends, and the water is pretty much just snowmelt there. If you're around camp on summit day, I wouldn't count on fishing nearby, although you might be able to hike back down a few miles to a lake or meadow. Your best bet for fishing would be to take a couple of extra days before or after the trip, maybe camp near the trailhead or something like that... you'll have plenty of spots to choose from.
Anyway that's just a few things off the top of my head. Hope it helps. Don't let anybody scare you away..