Learn English – “in the morning” vs. “in morning”

It is grammatically correct, but illogical.

"Any time during the morning" means "between midnight and noon" but "until 4pm" means, well, "until 4pm."

So to say "In the morning until 4pm" doesn't really make sense. Although most anyone will understand that you mean "In the morning or in the afternoon until 4pm."

I would simply say:

I am available any time until 4pm.

There is no reason to specify morning at all in your case.

First off, prepositions are tricky words, and it's hard to talk about all the ways they can be used.

That said...

at

We can use at night to mean during the nighttime:

I was there at night.

(notice how I did not say "at the night," which is what you wrote in your question).

in

We can use in the night to mean during the nighttime or at various times during the night:

Hyenas prowl in the night. (at night would work, too)

into

We use into the night to say that something continues for a long time after sunset:

The partygoers reveled late into the night.

on

You are correct, we don't usually say on night or on the night. However, if we expand the phrase, and stipulate which nights we are talking about, then "on (the) nights when..." is a relatively common construct:

The band liked crashing at our place on nights we played at Live Bay.
I try to go to bed early on Sunday nights.
On the nights when we hear the ice cream truck, we'll go outside and buy ice cream.

Now, let's combine some of these together:

We usually lock the front door at night. However, our roommate Micheal is an actor, and he sometimes works late into the night. On the nights Micheal hasn't come home before midnight, we leave the front door unlocked, in case he has forgotten his key.