I havent seen this show, but I work in pr, so i know something
Though I do not work in a pr company, I work as a pr person directly to one brand in entertainment industry.
For me, the biggest problem is that its impossible to control everything, often there are many things that just doesnt depend on you - though you are in charge of them. For example: you cannt control everything some journalist is going to write about you, or you cannt do anything if someone you are presenting wants everybody to think he's perfect but when he's in public he talks and acts like stupid...
Talking bout skills - you have to be energetic, very driven, not lazy, able to multitask and most of all - you have to go well with other people. This is the most important thing as your work depends on your contacts. You have to be always friendly and the "blah-blah" with everyone type.
Some people working in pr say that the fake friendliness pisses them off, but as for me - I like people, so I cant say that im so very fake.
And of course you need to know the right people, but this circle is built within a time period, so you could start firstly as an assitant and then, day by day, you would meet new people
Education wise - my bac is in law, but im working in pr. Though it helped me that I have worked for several years as a journalist when I was younger. Of course, studying pr is good, but I think that it would be even better to come from the journalistic background - one reasons is that pr includes a lot of writing and the other reason is that this there you would make your contacts! I havent studied pr, but everytinhing I know came to me naturally (as I have allways been more or less interested in this, even before understanding what it is), or from practise or from self educating - I was studying this a bit on my own.
Good luck!