Learn Hardcore Indonesian: Belakangan saya sibuk, tetapi umur tidak menghalangi untuk belajar, apalagi kalau materi diunggah cepat. - Lately I am busy, but age doesn’t prevent studying, especially if the material is uploaded quickly.

Belakangan here means lately/recently. By itself it’s natural in speech, but it can also mean later/afterward in other contexts.
Belakangan ini is the most common way to say lately and removes ambiguity.
Akhir-akhir ini also means lately (neutral).
Baru-baru ini is recently for discrete events, not ongoing states as naturally.

Examples:

Belakangan ini saya sibuk.
Akhir-akhir ini saya sibuk.

Do I need lagi or sedang with sibuk?

Not required. Belakangan (ini) already gives the time frame. Adding them changes nuance:

Belakangan ini saya sibuk. (natural, general state)
Belakangan ini saya lagi sibuk. (colloquial, ongoing)
Sekarang saya sedang sibuk. (neutral–formal, ongoing)

Should I use saya or aku?

saya: neutral/polite, safe in most contexts.
aku: casual/intimate, with friends or peers.
Choose based on your relationship and formality.

What’s the difference between tetapi, tapi, and namun? And is the comma before them necessary?

tetapi: standard but slightly formal.
tapi: informal.
namun: more formal, works like however at the start of a clause. Punctuation:
..., tetapi ... and ..., tapi ... take a comma.
Namun often follows a period or semicolon: ... Namun, ... or ...; namun, ...

Why umur? Could I use usia?

Both mean age.

umur: common, everyday.
usia: more formal/official. Examples:
Umur tidak menghalangi... (everyday)
Usia bukan penghalang... (formal)

Is menghalangi untuk + verb correct, or should there be an object?

Menghalangi is typically transitive. Many speakers prefer an explicit object or a rephrase:

Umur tidak menghalangi siapa pun untuk belajar.
Umur tidak menghalangi belajar. (drop untuk)
Very natural nominal rephrase: Umur bukan halangan/penghalang untuk belajar.

Is bukan halangan a better choice than tidak menghalangi?

Often yes. Indonesian likes nominal patterns here:

Umur bukan halangan/penghalang untuk belajar. It sounds smooth and idiomatic.

What’s the difference between untuk and buat in untuk belajar?

untuk: neutral–formal (preferred in writing).
buat: colloquial (common in speech). Both can mean for/to in this purpose sense.

Does apalagi mean especially or let alone?

Both, depending on context:

After a positive idea: apalagi = especially/even more so.
Example: Semua orang bisa belajar, apalagi kalau materinya menarik.
After a negative idea: apalagi = let alone.
Example: Saya tidak sempat makan, apalagi tidur. For unambiguous especially, you can use terlebih (lagi).

Is kalau okay for if here, or should it be jika/bila/apabila?

kalau: everyday spoken if/when; perfectly fine here.
jika/bila/apabila: more formal; use in writing or formal contexts. Meaning doesn’t change.

Does materi on its own mean learning materials, or should it be materi pelajaran?

In an educational context, materi is understood as course/lesson material. To be explicit or formal:

materi pelajaran, materi pembelajaran (more formal). Use bahan for physical materials (ingredients, supplies).

Is diunggah cepat natural, or should it be diunggah dengan cepat or segera diunggah?

All are possible with register nuances:

diunggah cepat: colloquial, concise.
diunggah dengan cepat: more formal/natural in writing.
segera diunggah: emphasizes promptness (soon/without delay), not speed per se. You can also say cepat diunggah or cepat-cepat diunggah (colloquial).

Why use the passive diunggah instead of active mengunggah?

Indonesian often uses passive to downplay or omit the agent:

Materi diunggah... (agent unspecified; general statement)
Active if you name the doer: Admin mengunggah materi dengan cepat.

Can I add -nya to materi?

Yes, -nya can mark definiteness or previously known items:

Materinya diunggah dengan cepat. (the materials) It can also be possessive from context: its/their materials.

Is the time expression position okay? Can I say Saya sibuk belakangan ini?

Yes. Both are natural:

Belakangan ini saya sibuk. (sets the time frame up front)
Saya sibuk belakangan ini. (focuses first on the state)

How would a more formal rewrite of the whole sentence look?

Belakangan ini saya sibuk, namun usia bukan penghalang untuk belajar, terlebih apabila materi diunggah dengan cepat.
Changes: belakangan ini (explicit), namun, usia, penghalang, terlebih, apabila, dengan cepat.