Are there any English sayings equivalent to the Japanese proverb, “Go to bed early and wait for the good news”?

I'd say something similar would be:

A watched pot never boils

Waiting for something to happen makes it seem like it is happening slower, whereas if you go away and do something else then time will seem to pass faster.

It's in God's hands now is something that English speaking people of faith often say in similar situations. I don't know if it counts as an idiom, however. It's more of an actual statement of faith, and as such wouldn't typically be used by people who weren't believers.

(As a comparison, there are also common English expressions like God only knows that are used simply as idioms, without implying any personal religious belief.)

There's no use worrying about it.

This is a common expression, although more literal and not really a "saying". Note that often it is shortened by omitting the "There's".

Those who are fatalists or polytheists might well say

Leave it in the lap of the g/Gods.

I would suggest "what's done, is done".

How now, my lord, why do you keep alone,
Of sorriest fancies your companions making,
Using those thoughts which should indeed have died
With them they think on? Things without all remedy
Should be without regard: what's done, is done.
Macbeth Act 3, scene 2, 8–12

While this may have a backward-looking sense similar to "no use crying over spilt milk" (i.e., one must let go of the past), it's often used with a mind to the now-impending consequences. A similar phase would be "the die is cast" (Alea iacta est, uttered upon Caesar's "crossing the Rubicon").