I cant speak to the Neem oil as I do not use it. But if you are spraying your trees to remedy or prevent a problem then yes you need to follow a spray schedule. Regardless if there is a chance of rain. Otherwise you lose some if not all the benefit of spraying.

What do you spray neem oil for? Where are you located?

If you are spraying as a preventative apply before the rain.

If you are spraying to eradicate a infection then apply after the rain.