I have a nasal irrigator (one of those that pulse water, like a water pick). I have used Lugol's in my solution many times. The water reservoir for it is 500 mL (very close to 2 cups), and I usually put about 8-12 drops in the solution (I always use horizontal drops, so that's a pretty high concentration), 1/2 tsp. Sea Salt , and 1/4 tsp. baking soda.
For a neti poti, you'll have to adjust these amounts to how much solution you are going to use.
Make SURE your water is close to body temperature, maybe slightly warmer. If your water is too cold or warm, it will sting a lot. I myself didn't notice too much stinging with the Lugol's (it did a little, but not too bad), but I have a pretty good tolerance for any Lugol's sting.
With a regular sinus irrigation without Lugol's, I use those measurements for the salt and baking soda. At those measurements, with the proper water temperature, it doesn't sting at all.
I recommend you first try a weaker solution with the Lugol's than I did, because some folks are more sensitive to it than me. Maybe start with 4 vertical drops. See how you handle it. Then if it doesn't bother you, the next time try a couple more drops. You'll find an amount that works well for you.
If you've a sinus infection, or bad allergies, you can flush your sinuses 2-3 times a day with the salt and baking soda solution. Usually when I do the Lugol's sinus flush I just do it once a day, but that's usually because of time restrictions and not because of any problems with the Lugol's. I did do it twice a day when I had a staph infection in my nose, and that killed the infection within a week (I'd had it for months). This will also clear your head nicely when you have a cold and help with congestion.