London for only 1 1/2 days.

It's something like 530 steps to the top gallery. Think of it as roughly equivalent to walking up the stairs of a 30-story building.
A good view, but a lot of work.
Standard advice for those wanting to do this is: first, go on up to the first viewing level, about half way. And then see how you're feeling. If you're knackered (as they say in the UK), or claustrophobic, or wary of heights--you can then decide whether to stop at the first viewing level, or continue the climb.
Note that it's a one way system, so you can't change your mind and turn around and go back down, until you reach the next level. A final long, tight, metal staircase takes you to the very top of the cupola.

If you haven't already done so, you might want to google the views from the top. [edit after reading subsequent posts], and compare those views to the views you could get from the London Eye across the River with less exhaustion.

On my first trip I had a humorous book called "Let's Blow Through Europe." In every city, it listed a couple of sites, bars, and a "tall thing to climb." But I myself never bother climbing tall things. The list of things to see in London seems endless.

With the amount of time you have, I wouldn't make the climb. If you really want a bird's-eye view, the London Eye shows you more for less effort. Might take a comparable amount of time but you're not wheezing at the end. Personally, with so little time I'd spend it all on the ground.

If you're interested in a behind the scenes tour at St Pauls, look at the Triforium tour. You'll get to see parts of it that aren't open to the public en masse- the library, the geometric staircase, the viewpoint from the camera mountings during televised events, and the original Christopher Wren great model. You're essentially climbing into St Pauls attic and you can decide if you want to continue on up into the dome afterwards. If you do go all the way up, it is a spectacular view if the weather is right and you can get some great photos.