It's possible to pass an amount to fill in the other field in the URL to a contribution page:
How can I pass in the contribution amount via URL?
But is it possible to pass a parameter in the URL to select other amount?
There doesn't seem to be any way to set other as the default even in the contribution page settings.
You can change the selected donation amount to any other option by passing the id of the option in the URL with the input name, but you can't seem to select the other amount option in the same way, as it is actually a different input element.