An equalizer with a high-shelf filter cutting higher frequencies will get you partway there. A touch of distortion or saturation on top of that should get you in the ballpark. You can play with the EQ frequency and gain cut to get the tone you like best and only use a bit of soft distortion, don't go overboard. This should be inserted after whatever normal mixing treatment needs to be applied to the voice over, such as normal EQ and compression.
I don't have any experience with STP and positively hate working with audio in FCP, so can't advise much on specific plugins to use. The default system AU plugins "AUHighShelfFilter" and "AUDistortion" should be able to get you there if that's all you have available. There are also more focused and professional (and expensive) tools, such as Audio Ease's Speakerphone, which is probably a bit much for your current needs.
It's an overdriven analogue recording. To match it you'll need:
Hard compression (10 or more in the ratio), or a limiter (drive the signal into it at high volume so that it's constantly reducing the level).
Distortion (just a little).
Probably a noise gate (or take a razor blade tool and cut out the quiet bits in between the words/sentences).
For the sections with reverb, I'd suggest a "plate" emulation to get that old style vibe.
A continuous loop of low level hiss would probably help as well.
Add the effects one at a time and adjust each one to find the "sweet spot" before you add the next one.