Shelf life of commercial jams with no preservatives? This commercial peach jam does NOT contain any preservatives.
Room temperature is 20 C.  

It is written on the bottle that it should be refrigerated after opening since it doesn't contain any preservatives.

For how long can I keep it in the cupboard after opening?
There'll be a lot of variation depending on temperature, humidity, the exact nature of the jam, and pure dumb luck, but I wouldn't be surprised if it started growing mold within a week, if not within a day or two. When things say to refrigerate after opening, they tend to mean it.