Most soda bread recipes I've come across (for example, this one) say something along the lines of "best eaten the day it's prepared", "will keep for a day" or similar. This is not the case for most sourdough or yeast based breads. So, why does soda bread preserve less well than other bread types? 
Almost all bread is best eaten on the day it is prepared, whether leavened with baking soda or with yeast.  (Exceptions might be some whole-grain dense breads with strong flavors that "settle" and improve after a day or so.)  Fermentation time does in fact have a small impact on how fast bread goes stale, though.  Staling generally encompasses two things: (1) degradation of the crust by excess moisture migrating from the interior of the loaf, and (2) hardening of the crumb due to recrystallization of certain starches.    The first aspect isn't that significant in soda bread, since its moisture content is high and its rising power (compared to yeasted bread) is relatively low, so it rarely develops the kind of crackly crust you might find on a French baguette.  In that respect, soda bread's crust won't degrade noticeably faster than, say, a typical moist yeasted sandwich bread.  The second aspect might make soda bread more likely to stale faster.  A long fermentation allows time for various enzymes to do some work in breaking down some starches in the bread and converting them to sugar.  If the starch content in the final bread is lower, staling will be reduced somewhat.  For example, amylase enzymes are naturally part of yeast fermentation and will break down starches, though sometimes they are added to improve bread texture and shelf life.  (Home bakers often use amylase in the form of barley malt.)  Is this effect significant, though, assuming you don't add enzymes to your bread dough during fermentation?  I'd guess that it probably only becomes significant for yeasted breads with long fermentation times, including those that use preferments or sourdough starters, or where fermentation takes place overnight in the refrigerator or something.  The long fermentation will allow time for the natural enzymes to activate and do their work.  So, some yeasted breads will have noticeable slower staling due to the process of yeast fermentation.  However, if you compared a soda bread recipe vs. a yeasted bread recipe with similar ingredients that is only fermented for a couple of hours, the difference in shelf-life between the two would probably be negligible. 