Пятнадцатого апреля у меня будет выходной.

Because Russian dates use an ordinal number, not a cardinal one.

• пятнадцать = fifteen
• пятнадцатый = fifteenth

So пятнадцатого апреля means the fifteenth of April / on April 15th.

This is very different from English, where we often say April fifteen with a cardinal number.

In dates, Russian normally uses the ordinal in the genitive form:

• первого января
• второго мая
• пятнадцатого апреля

Historically, this comes from an understood word like числа (date/day of the month), so the phrase is basically shorthand for something like пятнадцатого числа апреля.

For a learner, the practical rule is simple:
when saying a calendar date, memorize the pattern ordinal-genitive + month-genitive.

Because after the day number, the month is also put in the genitive:

• пятого марта
• десятого июня
• пятнадцатого апреля

So апреля is just the genitive form of апрель.

Russian usually does not use a preposition here. A bare date expression is enough:

• Пятнадцатого апреля... = On April 15th...

That is normal Russian. English needs on, but Russian usually doesn’t in this kind of sentence.

Literally, у меня means at me. In Russian, possession is often expressed with:

• у + genitive pronoun/noun
• plus a form of быть if needed

So:

• у меня есть книга = I have a book
• у меня будет выходной = I will have a day off

Russian often uses this structure instead of a direct equivalent of English have.

Because у requires the genitive case.

So the pronoun changes:

• я → меня
• ты → тебя
• он → него
• она → неё
• мы → нас

That is why you get у меня.

Because будет agrees with the thing that is being possessed: выходной.

Grammatically, the sentence is closer to:

• At me there will be a day off

The verb does not agree with я here, because я is not the grammatical subject.
The thing that will exist is выходной, which is singular, so Russian uses будет.

It is originally an adjective, but here it is being used like a noun.

The full phrase is really:

• выходной день = a day off

But Russian very often drops день and just says:

• выходной

So in this sentence, выходной means day off.

Not in the singular.

• выходной = a day off
• выходные = days off / the weekend

So у меня будет выходной means one day off, not the whole weekend.

Yes. Russian word order is flexible.

This sentence starts with the date because it sets the time frame first:

• Пятнадцатого апреля у меня будет выходной.

But you could also say:

• У меня будет выходной пятнадцатого апреля.

Both are correct. The version with the date first sounds a bit more like As for April 15th, I’ll have a day off.

Yes, in many contexts you can.

• Пятнадцатого апреля у меня выходной
• Пятнадцатого апреля у меня будет выходной

Both can be natural. The version with будет makes the future meaning more explicit.
The version without it can sound like stating a schedule or arrangement: I’m off on April 15th.

Not exactly. In standard pronunciation, adjective endings written -ого and -его are usually pronounced more like -ово and -ево.

So пятнадцатого is pronounced roughly like:

• pyit-náht-tsuh-tuh-vuh

You do not normally pronounce the last part exactly as written -ogo. This is a very common spelling-pronunciation mismatch in Russian.