I have a Brown Turkey fig. The rust on the leaves are common. I consider it a nuisance and all I do is remove the leaves. As long as the tree is otherwise healthy, I don't worry about it. Good sanitation helps. It cannot be stopped because the spores are in the air an soil and the fig is not immune. If you want to spray a fungicide, you need to spray when the weather conditions support the fungal growth and before the leaves get infected. If the tree is old enough to produce and it isn't, your tree either is getting too much nitrogen that keeps it vegetative growth or it is being stressed some other way. Your neighbors trees are producing.
If the fig is not producing, I would get a cutting from your neighbor's tree if the fruit is good and start over. Figs will fruit even in a 1 gallon can in a short time. I like Brown Turkey and Black Mission best. My friend had a Kadota fig that even the birds did not like, so some trees are just not going to have good fruit.
Figs like to be in full sun. They do like a good long drink but will not tolerate "wet feet". Deep watering once or twice a week if it is windy and really dry. If you want your fig to grow more laterally (like a bush) then use a lateral branch cutting (it has a memory). Figs too tall, cut it, it tolerates pruning well at the right time of the year, just don't prune any tree more than 1/3 at a time or it will be mad at you. Figs fruit on new wood, so don't prune off your fruit at the wrong time. If you are fertilizing the grass and you have a lot of leaves, it may be getting too much nitrogen.
Put the fig in a raised planter bed or a tree pot (25 gallons or bigger). I have a fig in a keystone planter 4 bricks high. We use leaf mold. We use leaf mold because someone said it was being used in Japan and might help keep the fig roots contained longer. P.S. the roots still broke out, but the fig really likes leaf mold. Otherwise it gets 10-20-20 a couple of times a year if I remember. The soil in this garden is poor in nitrogen. Even when the tree does not get water for a while it still will fruit, but the fruit will be very dry and the fig will lose leaves. My particular plot where I have the fig has phytophthora and it is only 17 ft above sea level. It is one of the reasons that the fig is in a raised planter and it will die if the roots go deep and hits the water table. The garden is in a basin and the herb garden floods in heavy storms.
https://www.hawaii247.com/2008/12/29/fa ... or-hawaii/
http://hihort.blogspot.com/2011/09/edib ... arica.html
https://aggie-horticulture.tamu.edu/ext ... /figs.html
Happy gardening in Hawaii. Gardens are where people grow.