Mihi hodie una pagina scribenda est, sed manus iam fessa est.

Because mihi is dative singular of ego, and this sentence uses a very common Latin pattern:

mihi ... scribenda est = I have ... to write / ... must be written by me

This is called the passive periphrastic with a dative of agent. Instead of saying I must write one page, Latin often says something more like:

one page is to-be-written for me / by me

So mihi does not mean a simple to me here in the usual English sense. It marks the person who has the obligation.

scribenda est is made of:

• scribenda = the gerundive of scribere (to write)
• est = is

Together they form the passive periphrastic, which expresses necessity, duty, or something that needs to be done.

So:

• pagina scribenda est = the page must be written
• mihi pagina scribenda est = I have to write the page

Literally, scribenda means something like to-be-written or needing to be written.

Because una pagina is the subject of scribenda est, so it is in the nominative case.

Even though in English we say I have to write one page, Latin structures it differently:

• English: I am doing the writing
• Latin here: one page is to be written by me

So pagina is not the direct object here. It is the thing that must be written, and therefore it stays nominative:

• una pagina = nominative singular
• not unam paginam, which would be accusative singular

Because it agrees with pagina.

In Latin, adjectives and adjective-like forms must agree with the noun they describe in:

• gender
• number
• case

Here:

• pagina is feminine
• singular
• nominative

So the gerundive must match it:

• scribenda

If the noun were masculine singular, you would get scribendus.
If it were neuter singular, scribendum.

Because manus is a fourth-declension noun, and in that declension some feminine nouns end in -us.

So although -us often suggests a masculine second-declension noun, it does not always do so. Manus is an important exception:

• manus = hand
• gender: feminine
• case here: nominative singular

That is why the adjective with it is also feminine: fessa.

Because fessa agrees with manus, and manus is feminine.

So:

• manus = feminine singular
• fessa = feminine singular adjective meaning tired

If the noun were masculine, you would use fessus.
If it were neuter, fessum.

Latin often leaves possessive words like my, your, and his unstated when the meaning is obvious from the context.

Here, if someone says:

sed manus iam fessa est

it is natural to understand:

but my hand is already tired

Latin could say manus mea, but it often does not need to. The context makes it clear whose hand is meant.

iam means already, by now, or sometimes now at this point.

So:

• manus iam fessa est = my hand is already tired

It suggests that the tiredness has set in before the task is finished, which fits the contrast with the first clause.

Yes. Latin word order is much freer than English word order because the endings show the grammatical roles.

This sentence is:

Mihi hodie una pagina scribenda est, sed manus iam fessa est.

A very literal breakdown is:

• Mihi = for me / by me
• hodie = today
• una pagina = one page
• scribenda est = must be written
• sed = but
• manus = hand
• iam = already
• fessa est = is tired

The order is natural and clear, but Latin could rearrange parts for emphasis. For example, putting hodie early emphasizes today, and placing iam before fessa highlights already tired.

Because there are two separate clauses, and each has its own verb:

1. Mihi hodie una pagina scribenda est
2. sed manus iam fessa est

In both clauses, the main verb is a form of esse (to be):

• scribenda est = must be written
• fessa est = is tired

Latin sometimes omits forms of esse in certain styles, but here including est in both clauses is perfectly normal and straightforward.

They are similar in meaning, but not exactly the same in structure or feel.

• mihi una pagina scribenda est = I have one page to write / one page must be written by me
• debeo unam paginam scribere = I ought to / must write one page

The first version is a very idiomatic Latin way to express obligation, especially when attention is on the task itself. The second uses debeo more like English must/ought with an infinitive.

Both can work, but the sentence you were given uses a very classical and common construction.