This is just a far fetched idea and I wouldn't know how to put it together, but you could have the tipping rain gauge activate some sort of counter, independent of the arduino, then when you need the rain fall info have the arduino poll the counter. This would assume that you only need the info once per day or some other predefined time period (this wouldn't help for IDF storm info).
If your displaying and collecting info somewhere downstream of the arduino, like a computer, and just have the arduino sending the computer the info, then using interrupts might work as long as you're not interrupting some time dependent function, and time stamp the info from the rain gauge on the computer.
I'm pretty new at this stuff so you might want to just take my ideas as conceptual, I really have no idea how you would go about it (although the tipping rain gauge, if it's just a simple mechanical one, you could add a magnet to it and use a reed switch to do the activation of when the gauge tips).