One of my teammates has died in battle. As a noble necromancer, I do not want to carry his corpse to the temple on my shoulders, instead I plan to animate his corpse, so he can walk on its own.

Then, in temple, I can destroy the created zombie and pay for his resurrection. Would he get resurrected properly?

Yes, you can resurrect him.

The resurrection spell says (emphasis mine):

You touch a dead creature that has been dead for no more than a century, that didn't die of old age, and that isn't undead. If the soul is free and willing the target returns to life with all its Hit Points.

Under normal circumstance I would argue that a destroyed/killed undead would no longer count as undead but Jeremy Crawford has posted a call on the similar revivify spell:

If you cast revivify on a zombie, the creature returns as a zombie. The spell doesn't change creature type.

- Jeremy Crawford

This call would let me extrapolate that it would not be possible. Except that this specific case is actually covered by the MM:

Once turned into a zombie, a creature can't be restored to life except by
powerful magic, such as a resurrection spell.
- MM pg. 315

This would mean the specific case of a resurrection spell (or if you want to be absolutely sure a true resurrection spell) it is able to turn a destroyed zombie back into the person it was before originally killed.

Though you might want to destroy the zombie outside of the temple and carry him the last part as I am guessing that most temples are not happy with zombies walking in.

Beyond RAW

This case relies on a lot of specific rules and exceptions if tis would come up at my home game I would allow resurrection of mindless destroyed undead like skeletons and zombies. I would not allow revivify even if the spell was cast before a minute has past since the original death.

No, you can't Resurrect your friend

Strictly by RAW, Animate Dead creates an undead creature, the thing is no longer "your friend's corpse", it is now a zombie with the zombie game statistics:

Animate Dead

Your spell imbues the target with a foul mimicry of life, raising it as an undead creature.

With the above, your teammate has become a zombie, and when destroyed, becomes a dead zombie and not a dead PC.

One of the prerequisites for Resurrection is that the corpse cannot be undead:

Resurrection

You touch a dead creature that has been dead for no more than a century, that didn't die of old age, and that isn't undead.

This is further supported by Jeremy Crawford's tweet:

If you cast revivify on a zombie, the creature returns as a zombie. The spell doesn't change creature type.

- Jeremy Crawford

You also cannot use Raise Dead on him as Raise Dead also states:

Raise Dead

The spell can't return an undead creature to life.

However, True Resurrection does not appear to have the same limitations as the previously mentioned spells, so you probably can get someone to cast True Resurrection, if you're able to pay.

Yes, the character can be resurrected.

Source, MM pg. 315 under Zombies, sub-section Dark Servants:

Once turned into a zombie, a creature can't be restored to life except by
powerful magic, such as a resurrection spell.

This is not specific to the spell Resurrection but rather resurrection magic in general. So killing the zombie and bringing the character back to life should be manageable within the rules of most spells so long as they qualify within the bounds of their own restrictions.

However, be careful on timing since the limits on resurrection magic start when the character dies. Animate Dead is not the same as Gentle Repose and does not extend the duration for which you are able to resurrect a dead character/npc. So for instance, Revivify only works if used within a minute of death. Raise Dead within the last 10 days.