Is the tree still reasonably alive? If at least 1/3 of the main branches are healthy, the possibility of rejuvenation exists.

It will take a lot of patience and hard work to rejuvenate an overgrown or neglected tree.
Before you proceed, ask yourself,, why are you saving this tree? Are you trying to save something special or are you doing it “because it is there?” If you like the fruit, or fear that you may lose the variety, you could gather some scion wood and graft new trees. The actual long term cost of a newly planted tree is often no more than rejuvenating an old tree, and requires a lot less care. Whatever you decide, I wish you luck. Slowly brining back a forgotten tree can be fun and rewarding!

As for pest control, Bayer makes a great product called Bayer Advanced Fruit, Citrus, & Vegetable control. It is a one year systemic mix-and-pour insect control. It is rainproof you can start using it any time. It is not organic.
If disease is your concern, the remedy and timing will be based on the problem and a little research is required before your proceed.
Pruning will benefit your tree in a number of ways, especially if infested or diseased. Don be afraid your cuts will do great damage to your tree, especially if it is on its last leg. Remember, should the renovation fail, it is likely the tree was beyond hope and not so much because of anything you did incorrectly.

Dan