Meaning-in-Context – What Do They Mean When They Say So?

It is indeed a rhetorical use of 'catch up with': a deliberately uncommon use, which attracts readers' attention so they will read the explanation.

The author contrasts the attitude he attributes to older people—that they are slaves to “long term plans that basically take a big chunk of your soul over a long period of servitude”—with the attitude of his own generation—“Life nowadays is INDEED about the NOW!“

His “catch up” figure thus represents the long-term plans of his elders as an unceasing effort to get ‘in front of’ who they are, while people of his own generation abandon that pursuit and turn back (so ‘back is the new front’) in order to ‘catch up with’ who they are.

"I'm partial to..." generally is used in the context of making a decision. It's equivalent to saying: "I'm biased towards (choice A)".

It can also be used for more general expressions of preference:

Friends is a good show, but I'm partial to Seinfeld.

The reason that it would be a joke in the first example is that using such formal, emotionless terminology about something that the speaker is supposedly very passionate about (their husband) is ironic.