Is it possible to get a day hike permit for either the main trail or North Fork on the same day as the planned hike. Ie. If I wanted to show up at the visitors center at opening time and get a same day permit just to do a partial hike (say up to trail camp or UBSL)?

Yep: pretty good odds in fact, especially weekdays.

I got a walk-in permit last Sunday for a Monday overnight, but they USFS rep said there were permits available for that same day, so yes.

What I could not do was get a walk-in permit for a hike 2 days in advance.

Also, walk-in permits are free. I didn't know that until this week.

The permits have always been free. What you pay for in the permit lottery is the lottery and reservation system.