It is easy to determine if you are running tubes.
Deflate the tire, leave the lock nut on the valve stem near the top. If the entire valve assembly want's to push into the rim, you have a tube. If the valve is fairly rigid, no matter the tire's pressure, you have a tubeless tire with a separate valve assembly.
You can perform a leak down test of sorts to make sure the the tire is holding air. Fill it to about 50psi at night and check the pressure in the morning. Given that you went from a great 50 mile ride to a squishy quarter mile, I'm willing to bet your tube/tire is compromised in some way. A visual inspection might help. Look for punctures, deep cracks, etc. If the tube is leaking, you could check for that by putting soapy water, window cleaner, or something else that bubbles around the valve stem hole in the rim, the valve itself, or the tire bead.
As said above, if the bike is new to you, you should really assess the condition of the tires. Keep in mind that old tires with good tread will crack along the sidewall pretty quickly. When I go my bike last year I rode it on the decent looking factory tires for about 1K miles. They aged very quickly! Fortunately the tubes looked great when I put in new tires and tubes. It was still a pretty stupid decision to ride them at all though...