How to keep celery crisp

Best thing to do is to keep them out of the light in an cool dark dry place. I usually put them in a hessian bag to try and keep the moisture away.

Don't store them in the fridge or anywhere that gets direct sunlight.

It should last years. If the reduction contains just vinegar, reduced until thick. This yields a very acidic solution very high in sugar (partly caramelized, depending on how long it was cooked). Other than sugar and acetic acid, there are some polyphenols and other gunk, some aromatics, and a bit of water.

There are two primary ways that food spoils: decay by microorganisms and oxidation. Decay happens when things get stinky and slimy, such as meat that has been left in a warm place for a few days. Bacteria can live in a lot of habitats, but need fairly mild conditions to thrive. Oxidation occurs when unsaturated fats (and other substances) pick up rogue chemicals from the environment, and is often refereed to as 'rancid'. This process will occur whenever something that can oxidize is exposed to air, but tends to progress slowly.

In the case of a balsamic reduction, the high acid and sugar content will stop bacterial growth. Since the reduction also contains acid, anything that could have been oxidized has already oxidized. Therefore a balsamic reduction should last pretty much forever. You might want to seal it up to keep bugs from getting in it, but other than that, it should be stable for years.