Я скачал его на ноутбук, но мне нужно установить его и на телефон.

Его here is the 3rd-person pronoun in the accusative (for animate/inanimate it looks the same as genitive in this form). It functions as the direct object of both verbs:

• скачал его = downloaded it
• установить его = to install it
  Repeating его is natural and clear, but in casual speech the second one can be omitted if it’s obvious: …но мне нужно установить и на телефон (still understandable, just slightly more elliptical).

Мне нужно literally means to me it is necessary/needed, and it’s the most common everyday way to say I need (to do something):

• мне нужно установить… = I need to install…
  Я нуждаюсь exists, but it’s more formal/bookish and typically used with nouns: я нуждаюсь в помощи (I need help). For “need to + verb,” мне нужно + infinitive is the standard choice.

With many devices and platforms, Russian often uses на to mean “onto/into the device/system” in a practical sense:

• скачать на ноутбук = download onto a laptop
• установить на телефон = install on a phone
  You may also hear в телефон in some contexts (especially with putting something physically “into” something, or with certain apps/files in casual speech), but for software installation/download destination, на + device is extremely common and idiomatic.

Both can translate as “download,” but usage differs:

• скачать is the most common everyday verb for downloading from the internet.
• загрузить can mean to load or to upload depending on context, and it’s broader/less specific.
  In this sentence, скачал is the most natural choice for “downloaded.”

Because the phrase shows direction/target of the action (where the file went). With на for direction you use the accusative: на ноутбук.

• с ноутбука would mean “from the laptop” (source).
• в ноутбуке would mean “in the laptop” (location), not the target of downloading.

Word order is fairly flexible, but it changes emphasis:

• Я скачал его на ноутбук… is neutral.
• Я его скачал на ноутбук… emphasizes it a bit more (topic-first).
• …но мне нужно… is the most natural; …но нужно мне… is possible but sounds marked/poetic or like special emphasis.

Его can refer to something masculine or neuter:

• файл (masc) → его
• приложение (neuter) → его
  If you meant программа (feminine), you would use её instead: Я скачал(а) её…. In practice, people often mean “the app/file” and его is very common because приложение/файл fit well.