Yes chapstick and lip balm does work good for putting inside your nose.
I use the medicated lip balm to help prevent irritation of my nose when it gets dry during the winter months or anytime the air is dry.
After using lip balm or chapstick inside my nose my nose doesn't get irritated like it used to during the dry weather and winter months.
You can also put Vaseline up your nose to prevent nosebleeds and irritation from dry air.