Vent cleaning/system clearing

[stern.ed]

I have a slow basement sink with a clear p-trap. Actually the standing water level is just at the top of the basin tailpiece.

Other drains in the house appear to work fine, including a basement toilet and sink. I wonder if it's a vent problem at/near the basin?
If so. what's the right way to check a home plumbing vent? I read about a garden hose as a pressurized way to clear a vent from the roof-down, or using said hose as a crude snake.

The plumbing itself on this basin is old. All bronze, likely original. It's right near the 4" main drain and trap. I'll post a photo in a few hours.

[Terry]

Most of the time, it's at the stopper. Remove the trip lever rod in the back of the drain assembly, pull out the stopper and get all the gunk off of it, and then run a hand snake right down the drain.

[stern.ed]

Hrm, no stopper-directly overboard. Here's a photo.

Edit: I thought that bulbous thing was a unique 't' joint. It's a check valve. Before I crack the seal, which is on with 20 years of natural-snugness, what is the right way to re-pack the gasket on the check valve? Is it older flax/grease packing?