Hello Alexander - welcome to the forum!
I'm not sure it is possible to clear the rain counter on WH1080 - certainly you can clear the total rain as displayed on the console but I don't think this will affect the total rain reported to the computer (willing to be corrected here - check with the manual that came with your station) The WH1080 only has one rain output - the total, so WeatherCat will take that as the daily rain when first run (if it's less than a few 10's of millimetres) - you can correct it with the daily rain editor tomorrow.
Be aware that if you've mounted the rain gauge on the mast, it may read rain when there is none as it's quite susceptible to vibrations - wind for example - I'd recommend mounting the rain gauge on something more solid that the mast that comes with the station.
Finally, you can clear all WeatherCat's data by going to its preferences (Location/Colours) and hitting the Delete Data button - this will clear everything apart from the settings. Once it's done, reboot WeatherCat.
Cheers,
Stu.