There's a copy of the song on archive.org: https://archive.org/details/my-neighbor-totoro-ost/01+Hey%2C+Let's+Go+(Opening+Theme).mp3

Edit: Found a second copy which might be slightly better quality -- https://archive.org/details/my-neighbor-totoro-cd/CD3+-+My+Neighbor+Totoro/01+sanpo+-+oopuningu+shudaika.mp3

Edit 2: Hmm. The version on my DVD doesn't seem to have that buzzy drum roll at the start (or at least, it's been mixed so that its way less noticeable) -- otherwise, it's very similar.

Edit 3: The DVD copy has an instrumental section after 1 minute, so while the opening of the song is basically the same, it's somewhat altered from these. I'll see if I can get a clip from it later.

Edit 4: Had some trouble with the DVD but it turns out I actually have the BD too. Had to dig it out of a box since it didn't play on my computer when I bought it ~10 years ago... but weirdly, that turned out to be easier to get a clip from now, so here's the OP from the BD instead for comparison: https://0807.st/p/aMPh2gm.mp3