Hún sagði að hún væri nýbúin að sjá fyrsta túlípanann opnast í morgun.

Because this is reported speech after Hún sagði að ....

Væri is the past subjunctive of vera. In Icelandic, the subjunctive is very commonly used in subordinate clauses that report what someone said, thought, believed, and so on.

So:

• Hún sagði að hún væri ... = She said that she was / had just ...
• A likely direct version would be Ég er nýbúin að ... = I have just ...

Using væri helps show that this is her statement being reported, not just the narrator’s own flat statement of fact.

Vera nýbúinn/nýbúin/nýbúið að + infinitive is a very common Icelandic expression meaning to have just done something.

So:

• hún væri nýbúin að sjá = she had just seen
• literally, it is something like she was newly finished with seeing

This construction strongly emphasizes recent completion.

Examples:

• Ég er nýbúinn að borða. = I’ve just eaten.
• Við erum nýbúin að koma heim. = We’ve just come home.

Because nýbúinn behaves like an adjective and agrees with the subject.

Here the subject is hún, so the form is feminine singular:

• masculine: nýbúinn
• feminine: nýbúin
• neuter: nýbúið

So:

• Hann er nýbúinn að lesa þetta.
• Hún er nýbúin að lesa þetta.
• Barnið er nýbúið að lesa þetta.

They do two different jobs.

1. Hún sagði að ...
   Here að means that and introduces a subordinate clause.

2. nýbúin að sjá
   Here að is the infinitive marker, like to in English.

So the sentence contains:

• að = that
• að sjá = to see

They look the same, but they are not the same grammatical item.

Because sjá takes a direct object, and here that object is in the accusative.

The noun is túlípani. In the singular definite accusative, it becomes túlípanann.

Also, the adjective fyrsti changes to fyrsta because:

• the noun is definite (the tulip)
• Icelandic adjectives before a definite noun usually take the weak form

So:

• nominative: fyrsti túlípaninn = the first tulip
• accusative: fyrsta túlípanann = the first tulip as an object

In this sentence, it is the object of sjá, so fyrsta túlípanann is required.

This is a common pattern after verbs of perception such as sjá (see) and heyra (hear).

The structure is:

see/hear + object + infinitive

So:

• sjá fyrsta túlípanann opnast = see the first tulip open

This is very similar to English:

• I saw the flower open
• She heard the door close

In Icelandic, opnast stays in the infinitive because it describes what she saw happening.

Because opnast is the natural verb for something opening by itself.

• opna usually means open something
  Example: Hún opnaði gluggann. = She opened the window.

• opnast means open / come open
  Example: Glugginn opnaðist. = The window opened.

A tulip is not normally something that someone actively opens; it opens on its own. So opnast is the idiomatic choice here.

Í morgun means this morning.

In this sentence, it most naturally refers to the event of her seeing the tulip open:

• She said that she had just seen the first tulip open this morning.

Because it comes at the end, it sounds like a time expression for the whole event. In real usage, listeners usually understand it without difficulty from context.

A likely direct version would be:

Ég er nýbúin að sjá fyrsta túlípanann opnast í morgun.

Then in indirect speech:

• ég becomes hún
• er becomes væri
• nýbúin stays feminine because the speaker being referred to is still female

So:

• direct: Ég er nýbúin ...
• indirect: Hún sagði að hún væri nýbúin ...

Yes. Since the subject of both clauses is the same person, Icelandic often uses segjast.

So a more compact alternative is:

Hún sagðist vera nýbúin að sjá fyrsta túlípanann opnast í morgun.

This is roughly equivalent to:

Hún sagði að hún væri nýbúin að sjá fyrsta túlípanann opnast í morgun.

The version with sagðist is very natural when the person who said something and the person in the subordinate meaning are the same.

You could express the idea in other ways, but vera nýbúin að + infinitive is especially good for have just done / had just done.

Compare:

• hafa séð = have seen / had seen
• vera nýbúin að sjá = have just seen / had just seen

So the sentence is not only saying that she had seen it; it is also emphasizing that this happened very recently. That is why nýbúin að is such a natural choice here.