A ponte velha precisa de reparação.

In Portuguese, precisar de is the standard verb-preposition combination used to express “to need.” You cannot say precisar alone with a direct object. You must include de before the thing that is needed.
Example:

• Eu preciso de um guarda-chuva. (I need an umbrella.)
• Ela precisa de paz. (She needs peace.)

Here reparação is used in an uncountable, general sense (“repair work” rather than one specific repair). In Portuguese, when you talk about something in a general, abstract or mass sense, you often drop the article:

• Precisa de reparação (needs repair in general)
  If you wanted to refer to a specific repair, you could use an article:
• Precisa de a reparação urgente (needs the urgent repair).

Portuguese generally places descriptive adjectives after the noun:

• ponte velha (old bridge)
  Putting velha before the noun (velha ponte) is also grammatically correct but can add stylistic emphasis or poetic flavor. In everyday speech, Post-noun placement is more neutral.

Yes. Both words can mean “repair,” but with slight differences in register:

• reparação tends to sound more formal or technical.
• conserto is more colloquial and common in spoken language.
  So you could say:
• A ponte velha precisa de conserto.
  and it would be perfectly natural.

Definitely. You can use the passive infinitive with ser + past participle:

• A ponte velha precisa de ser reparada.
  This literally means “The old bridge needs to be repaired.”
  Note the agreement: reparada matches ponte (feminine singular).

In Portuguese, most nouns ending in -ção are feminine. You can see the gender by the article or adjective:

• a reparação (feminine)
• uma construção (feminine)
  Knowing this pattern helps you pick a vs. o, uma vs. um, and agreement with adjectives.

Both express necessity but in slightly different contexts:

• ter de
  • infinitive indicates obligation or duty (must, have to):
    • Tenho de estudar para o exame. (I have to study for the exam.)
• precisar de
  • noun indicates need or lack of something:
    • Preciso de um lápis. (I need a pencil.)
    You wouldn’t say tenho de um lápis or preciso estudar without de (unless you use precisar + infinitive, which means “to need to do something”: Preciso estudar = “I need to study”).