That doesn't come across as authentic, that comes across as transactional and having an agenda.
"Hey, it's been forever, want to catch up over some food?"
If you end up hearing about a useful opportunity, or providing them with a useful opportunity, great. But you still got to catch up with a friend, and came across as that being the primary goal.
If you are looking for part-time work, it's fine to say that, and there's nothing wrong with reaching out to friends to find out if they know of good job opportunities. But "share what we're both working on and how we might help each other?". That comes across terribly.
In my experience, it's a coin toss on whether having the last bit increases or decreases a chance of the appointment being made. Some people require it.
Ultimately the only thing that matters is: you have a phrase that you can say comfortably and expresses genuine curiosity about the other person.
"Hey, it's been forever, want to catch up over some food?"
If you end up hearing about a useful opportunity, or providing them with a useful opportunity, great. But you still got to catch up with a friend, and came across as that being the primary goal.
If you are looking for part-time work, it's fine to say that, and there's nothing wrong with reaching out to friends to find out if they know of good job opportunities. But "share what we're both working on and how we might help each other?". That comes across terribly.