What's a good strategy for getting the Jump-Rope Genius moon?

An important thing to note is that the jump rope will keep speeding up until it hits 50 jumps. Once you get to 50 jumps, the rope will continue moving at the same pace indefinitely.

Here are some tips from gosunoob which I found fairly helpful. (Bolded the tips I found most useful)

Metro Kingdom Jump-Rope Challenge Tips

• The harder you press the button, the higher Mario jumps. Tap the button as lightly as possible – minimal jump height will ensure you don’t spend too long in the air, which is crucial after the speed ramps up.
• Try different camera angles. Keeping the camera behind one of the women worked best for us, but you might feel more comfortable with another setup.
• Stand halfway between the women, as this will give you the most leeway.
• Once you pass 50 jumps, stop looking at the rope. Instead, keep an eye on Mario’s feet, and jump as soon as he hits the ground.
• Up to 30-40, you can use the “Hey” shout as a timer.
• Some people think the lag from the joycons is an issue. If you have the pro controller, try using that instead. Alternatively, try the handheld mode.
• Riding the moped/motorcycle might make things much easier, but it probably depends on the person. You can find it parked by the curb nearby. Align yourself with the rope, then wait for a good opportunity to just ride into the game.
• You can activate two-player mode in order to set the camera in a more fixed position, then leave Cappy by the side while Mario jumps.

You should be able to reach 50 jumps after some practice.

Once you hit 50 jumps, as Steven Vascellaro said in his answer, the pace will stay constant.

At this point, stop trying to time your jumps, and mash your jump button as fast as possible. This will cause Mario to do the smallest jumps and also to jump as soon as he lands.

This does not always work (I've had it fail around 70s and 80s), but after a few tries you should reach 100. More importantly, since you don't have to concentrate after 50, it will keep you from losing your mind.

Although I haven't gotten 100 yet (51), I have found that it's easier if you do short jumps all the way through. As tempting as it may be to do longer jumps earlier on as the rope is moving slower, it is counter-intuitive to do this as you will have a harder time mentally switching gears into short jumps around the 50 mark if you've been doing tall jumps up to this point. Just get into the habit of doing all short jumps by 35 or 40, and you should find the right groove more easily. Best of luck!