You need to have an experienced shooting instructor take a look at your technique. What you're complaining of is fatigue. In addition to strength issues, fatigue can be caused by overcompensating for a control problem by squeezing the gun too firmly. I learned this because I have a congenital problem in my wrists called collateral ligament laxity, or commonly referred to as "limp wrist". No, not that kind of limp wrist...
Basically, I am either missing ligaments in my wrist that give support to the joint, or they are very lax and do not provide good support to the wrist joint. You can actually see this by fixating my hand then moving my forearm, the bones in the wrist "float" a lot more than they should. Kinda hard to imagine it unless you see it, but everyone thinks its 'gross' when I do it. haha
The result of this is that when I fire a handgun, there is an inordinate amount of play in the wrist joint, causing movement and articulations in the joint that shouldn't happen. So the energy from the recoil is not transfered well through the wrist joint into my forearm and upper body. This kills my control and invariably makes me "squeeze" the dickens out of the frame to compensate. It also causes quite a bit of pain as my carpal and wrist bones 'slam' together during recoil. Because I'm gripping the frame like nobody's business, my arms and hands get fatigued fast.
It took me forever to figure out why people with half my upper body strength were better shooters, had better control than me, and could shoot longer without fatigue. I happened to shake hands one day with an orthopedic surgeon and he felt my wrist bones 'clunk' together loudly. The 'clunk' transferred through his hand. He was like, "What the hell was that?" I shrugged my shoulders and said my wrists always do that.
So he picks my wrist up and starts manipulating it. After about ten seconds he was like, "Hey this is cool! I haven't seen this since I was a resident." I'm like, you haven't seen what? So then he starts explaining this ligament condition and has to invite two other orthopedic residents to come play around with my wrists.
Anyway, I asked him how that would affect my shooting, and he said it would probably make it very difficult if not painful. I was like..."ah-ha!" (light bulb over my head lights up).
Lifting weights will help if your fatigue is caused in any significant way by strength issues. Exercises that strengthen the upper body particularly your shoulders, forearms and pecks, should help.