Question
This summer my mother bought me a brown turkey fig tree. We
kept it in its original pot (2.3 qt) in a gated garden. It
had plenty of sunlight and it was watered regularly, but
the leaves were turning yellow. Before I moved to college
we transplanted the plant into a larger pot, and now I
continue to care for it.
The plant is about a foot and a half tall, with two
prominent branches. It has 5 fruits, one of which is
ripening. Some of the yellowing leaves fell off, but there
are still two remaining that are turning brown at the
edges. The other leaves are green, but some of them are
turning brown at the edges as well - 3 of them have brown
holes on the leaves.

What should I do about the leaves on my plant?
When is the best time to pick the fruit from my plant - to
ensure that the fruit will continue to grow back, and grow
back larger?  

Answer
There are three things to consider when looking at your trees yellow leaves.
The first has to do with watering. Fig trees in containers are seriously fussy about having moist, well drained soil. Commercial growers add a little sand to woody container soil, to retain the moisture and still keep the water drainage. You can also add a moisture polymer to your soil. I buy potting soil with a polymer already in it. Water often, especially in hot weather. If it stays hot for many days, move the tree to the shade, until it cools again. These trees need at least 8 hours of sun a day.
The second is mealy bugs, scale, or aphids. The holes in the leaves may be a clue to look for one of these pest. The most common would be aphids. If there is a sticky residue on the leaves or stems, and if you see ants, then the problem is aphids. Clean the leaves and spray with a pyrethrum based insecticide.
The third is a fungal disease characterized by small yellow green spots that eventually turn brown. This disease is triggered by excessive wet weather and causes the leaves to drop prematurely. Remove all of the diseased leaves. If the weather remains wet and the disease persists, spray with copper-sulfate fungicide such as Bordeaux mixture.
Since I can't personally look at the tree to see what is wrong, I have given you some things to look for. As you can see, more then one thing can cause the leaves to yellow and fall off. But, if you look back at how the tree was planted and cared for, and how the climate has been, you will probably find the answer to your question. It can be more then one thing wrong, as well.
As for the answer to your question about picking the fruit. You live in a cold area in upstate New York. You have a short growing season. Unless you have a warm greenhouse for your tree, your fruit may not have time to ripen on the tree. This means that there may be fruit that never has time to ripen before the tree goes dormant for the winter. This is a Mediterranean tree. It grows naturally in a mild, warm climate. So to answer your question, you will need to pick as many fruit that can get ripe in your summer climate, and enjoy what you have.
I would suggest finding some good web sites that will give you detailed information on all of the things you need to know about caring for your tree. You will need to know how to pot up the tree, water it, fertilize it, prune it, over winter it, and harvest it.
Once you have mastered these things, you will experience fewer problems.