I have noticed that I need to urinate more often than before when I wear my menstrual cup. Is this normal?

Sometimes the cup increases the need of urinate more often if the vagina is tilted towards bladder. This means that the rim of the cup pressures the bladder which cause need to pee more often.

Try to place the cup a bit higher or lower to find a spot where it's comfortable. Size 1 is softer than size 2 and might suit you better.