Well, about building upper body strength on a bike, you ride a bike you don't carry it. Climbing hills is all about power to weight ratio. Light, aerobically fit riders make good hill climbers. Big riders do not.

As an illustration, I recall one ride. I came across a couple of ladies. We were riding the same route (which was rather hilly). I offered to pull them. After a bit, another rider came along. He was very fit, lean and big from all of his muscles. He wanted me to go ahead with him and drop the ladies. I told him that I was going to stick with them. He would go ahead some of the time. On one of his breaks, I told the ladies that we would drop him at the third big hill. The ladies were fit and thin. The great big strong guy just could not hang with them on the hills. He was off the back.

Also, I do not think that weight lifting helps riding much. Weight lifting tends to be an anaerobic activity. Cycling tends to be aerobic. Also, it is very difficult to mimic the cycling motion by lifting weights.

Riding out of the saddle with a high intensity on a spin bike would help hill climbing assuming you don't have hills to practice on. Nothing is better to practice climbing than climbing. Another thing would be working out on a stair climbing machine. Also, a third thing would be losing weight. Getting your body weight reasonably low will improve hill climbing assuming you don't do it too quickly and eliminate muscle in the process.

Finally, you might need lower gearing. I like high RPMs so I tend to take hills in lower gear than other riders. I climb just as fast and often faster but I am still in a lower gear.

Carbonfiberboy has a point. If you are riding and you hit a really, really steep hill, you can walk it. It isn't illegal. Also walking the hill is better than burning up your muscles and seriously impairing your ability to ride the rest of the ride. I have not had to do it. I can do a 20% grade as long as it is not too long (and I have never seen a long one). I have also never had the opportunity to climb anything steeper than 20%.