The best and most straightforward strategy is to simply upgrade the Jackdaw. We're not talking minor hull plating here or a few cannon upgrades. If you maximize your upgrades, there are few ships that will be able to match your firepower and defensive capabilities. The only thing you'd have to worry about are ships outrunning the Jackdaw, and that's something that can be taken care of with a good battle strategy.
When you're going up against higher level ships, offensive firepower and defensive hull plating are equally important. Even if you have elite weaponry, it won't matter if a single mortar shot wipes out your health bar. Likewise, if you have enough hull plating to withstand a barrage of attacks from a level 60 vessel, but can't make a dent in its hull, you won't be winning that battle.
Max Range Mortars
While most of your battles against high level ships will take place at close range to avoid mortar fire from your opponents, it's still beneficial to upgrade your own mortar attack. You can hit an enemy ship with mortar shots long before you can engage them with any other form of attack. This gives you the jump and allows you to approach the enemy on your terms. Keep in mind, once you attack, they will fire their own mortar shots, so be ready to dodge.
The best way to avoid an enemy mortar is to engage at full speed. As soon as you see the yellow circles indicating a mortar attack is coming, slow down as much as possible. In many cases, an enemy mortar attack is aimed to hit your ship in the position it will be in when the shots hit the water, given your present speed. If you slow down, and even throw in a sharp turn, it's relatively easy to avoid these attacks.
Aim for the Back
The higher level ships tend to have very solid hull plating. However, this hull plating doesn't generally cover the front or back of the ship. It's almost always focused on the sides of the ship. For this reason, aim your broadside cannons at the back of the ship to inflict the most damage. You can also use a strategy of ramming the ship in conjunction with chain guns (also targeting the back of the ship).
It's highly recommended to aim low for the best results, but even if you aim low on the sides of the enemy ship, you won't be inflicting much damage. The problem is maneuvering around the ship to target the back or front. If you're behind the ship, but too far away, you may have to deal with explosive barrels. Keep this in mind as you approach the target, and try to come in at an angle so you avoid the broadside cannons and explosive barrels.
Learn to Swim
Possibly the easiest, but most long-winded strategy is to simply stop your ship just outside of the Man o' War's range, then swim to the ship. You'll have to use stealth swim when you get close in order to avoid getting shot in the water. Once you're on board, kill all of the crew members, then return to the Jackdaw. With the entire crew dead, a single shot from the Jackdaw will incapacitate the ship.
The problem with this strategy is that you have to start swimming from a considerable distance. You can only swim so fast, which means it takes some time to get to your target. In addition, depending on your angle of approach, you may have to avoid gunfire from the crew of the ship at quite a distance. The time you can spend stealth swimming is also limited, so you need to be able to maneuver around the gunfire and avoid taking hits until you're close enough to board the ship.
Despite the drawbacks, this is still the easiest way to take down a Man o' War, and it doesn't require you to upgrade the Jackdaw. If you have a lot of time on your hands, this could be your best option. Of course, if you have a lot of time you have time to upgrade the Jackdaw and fight the real way too...