Frame size doesn't really matter.
11-32 does make a substantial difference; remember its the ratio of the front gear to the rear gear that matters. Your lowest gear will be 12.5% lower. Also, you'll get a wider (and likely more useful) spacing in gearing between changes.
You will need a new chain as well, but it should be fine to install. Putting a bigger cassette on is a good starting point.
Of course, you have to shift appropriately -- if you're running 50/11 all the time, you're going to be tired regardless of what the biggest cog is. A 11 small cog is not very useful all the time.
Finally, "extremely steep" has a different meaning to everyone and their abilities (and the type of biking they're doing). For example, a touring bike in the mountains might be using a chainring in the 20's of teeth and some huge cassette to get the gearing needed to comfortably ride up the mountains. Most racing bikes aren't really built with that kind of riding in mind remotely.
Also, if you're using a 47 cm frame and it has 700c wheels, you can also get a reduction in gearing (and height) by switching to 650b wheels (easy if you have disc brakes; bit more work if you have rim brakes, but might be possible). My personal view is people below 5'5" ish shouldn't be riding 700c wheels.