Learn Hardcore Arabic (Egyptian): انا محتاج اشتري خضار من السوبرماركت قبل ما ارجع البيت. - I need to buy vegetables from the supermarket before I go back home.

Yes. محتاج literally means needing / in need of, but in Egyptian Arabic it is very commonly used to mean I need to... when followed by a verb.

So:

أنا محتاج أشتري = I need to buy

English uses to before buy, but Egyptian Arabic usually just puts the verb directly after محتاج.

Also, محتاج agrees with the speaker’s gender:

male speaker: أنا محتاج
female speaker: أنا محتاجة

Why is it written اشتري and ارجع without أ at the beginning?

That is very common in casual Egyptian writing. In more careful spelling, many people would write:

أشتري
أرجع

But in texting, chat, subtitles, and informal writing, the hamza is often left out:

اشتري
ارجع

So this is mostly a spelling style issue, not a meaning difference.

Is أنا necessary here, or can I drop it?

You can often drop أنا in conversation if the context is clear:

محتاج أشتري خضار...

That said, keeping أنا is very natural and helpful, especially for learners.

Unlike a fully conjugated verb, محتاج by itself does not clearly show person, so أنا makes the subject explicit.