Sounds like "loneliness is on the rise" which would mean that the amount of people feeling lonely is increasing. But yeah, it doesn't sound like is well phrased.

That's bad English in my opinion. It doesn't mean anything to me.

There's a casual phrase "on the up and up" that means (1) honest, and (2) on the rise. Maybe the writer meant to say "on the up and up" and have it mean definition (2). But even so, I think it's still a bad application of the phrase because "on the up and up" conveys a sense that the situation is improving. So in this case, it would be ambiguous. Like Alejo said, I'm pretty sure the writer wanted "on the up" to mean "on the rise" or "increasing." Overall this is just a bad sentence because it uses "lonely" and its derivative in one short sentence. If you take out the middle part, it's basically saying "anyone can feel lonely when their loneliness increases."