First off, how do you store your yeast? Storing yeast in the fridge helps it to last longer; I've had some yeast in my fridge for a year that is still going strong.
As your yeast begins to become ineffective, I would personally get new yeast. Once yeast looses its power, you're simply not going to get the same effect out of it.
As eggs are a very versatile ingredient: when substituting, it is important to determine whether the eggs are being used as a leavening agent or binder (or both). In recipes where the egg is used primarily as a leavening agent, I have used a mixture of baking powder (not soda), water, and vegetable oil in a 2:3:3 ratio, though you may want to experiment with the proportions depending upon your recipe. You may also need to vary the amount of other liquid in the recipe.
Where the egg is used primarily as a binder, you could use banana, applesauce, or gelatin, depending upon the recipe (obviously, you'll probably want to avoid the fruits in a savory recipe). I've used, and been thoroughly disappointed with, the available commercial "egg replacers" and don't recommend them for any purpose. I've heard of using flax, but have no personal experience with it.
However: as the father of (and resident chef for) a young child with egg and milk allergy, it is my experience that successfully substituting for eggs is very difficult. For breads in particular, I don't even bother with recipes that call for eggs or milk, and favor those that use the basic 3 (4): flour, water, yeast (salt).
Best Answer
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.