Вчера мне нужно было встретить сантехника, но я задержалась на работе.

Мне нужно было + infinitive is a very common, neutral way to say I needed to / had to. It’s an impersonal construction: literally something like to me it was necessary.
Я должен/должна был(а) also means had to, but it can sound more like duty/obligation (or a stronger “I was supposed to”), and it must agree in gender/number with the speaker.

Мне is dative case of я. In the pattern мне нужно/надо/можно/нельзя, the person who “has the need/permission” is put in the dative:

• Мне нужно было… = I needed to…
  Not меня (accusative/genitive) and not я (nominative), because the sentence isn’t built as “I + verb”; it’s built as “to me + it was necessary”.

Нужно is a “category of state” word (similar to можно, нельзя, надо) and in the past it typically uses было in neuter singular as a default impersonal past form:

• Present: мне нужно
• Past: мне нужно было
  It doesn’t agree with мне (dative doesn’t control agreement). You can get agreement when нужен/нужна/нужно/нужны is used as a short adjective with a real subject, e.g. Мне нужна помощь (help = feminine subject).

After мне нужно было, Russian normally uses an infinitive to name the required action:

• мне нужно было сделать / позвонить / встретить…
  So встретить is simply the action that was necessary.

Встретить is perfective, used for a single, completed event (meet someone once, catch them when they arrive).
Встречать (imperfective) would fit if you mean a process/habit/repeated action or focusing on the act of meeting as an ongoing activity (e.g. “I was meeting/used to meet people” or “I was in the middle of meeting”). Here it’s one concrete appointment, so встретить is natural.

Because сантехник is animate masculine, and in the accusative singular animate masculine nouns usually take the genitive form:

• Nominative: сантехник
• Genitive: сантехника
• Accusative (animate): сантехника
  So встретить сантехника = “to meet the plumber.”

Yes, but it changes the structure slightly:

• встретить сантехника = meet the plumber (direct object, accusative)
• встретиться с сантехником = meet up with the plumber (reflexive verb + с + instrumental)
  Both are common; встретиться often implies a planned meeting between two parties.

Past tense verbs in Russian agree with the subject in gender (singular):

• Female speaker: я задержалась
• Male speaker: я задержался
• Plural: мы задержались
  So the form tells you the speaker is female (or the “I” refers to a woman).

Задержаться на работе means to stay late at work / be held up at work (you remained longer than planned).
Опоздать means to be late (arrive late somewhere). You could be late because you задержались, but the verbs describe different things:

• Я задержалась на работе = I stayed late at work.
• Я опоздала = I arrived late (to a meeting, to home, etc.).

For “at work / at the workplace,” Russian normally uses на + prepositional: на работе.
В работе is used in other meanings like “in (the process of) work,” “in a piece of work,” or “in operation,” depending on context.

Because но connects two independent clauses with different subjects/verbs:
1) Вчера мне нужно было встретить сантехника
2) я задержалась на работе
In Russian, a comma is standard before но in this kind of compound sentence.

It’s flexible. The given order is neutral and natural, but you can move parts for emphasis. For example:

• Мне вчера нужно было встретить сантехника, но я задержалась на работе. (emphasizes to me / the obligation)
• Вчера мне нужно было встретить сантехника, но на работе я задержалась. (emphasizes at work)
  The grammar stays the same; word order mostly changes emphasis and style.

Common stress patterns:

• вчерА
• нУжно
• былО
• встретИть
• сантЕхника
• задержалАсь
• рабОте (in на рабОте)