Can I kill Lurrus directly after it enters the battlefield, before it's controller gets priority to cast a permanent from their graveyard?
Not normally. When ever a spell resolves, per the rules of magic, the active player gets priority.

117.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.

This means that your opponent who cast Lurrus will almost always have the opportunity to cast a spell before you get priority to try and remove Lurrus, in the same way players pretty much always have the ability to use a planeswalker ability before their opponents can kill the planeswalker.
There are three edge cases I can think of that would make this untrue:

Lurrus entering the battlefield caused some ability to trigger, say Soul Warden. This means that something goes on the stack before Lurrus' caster has priority, and since the spells Lurrus allows to be cast are generally not instant speed, the caster will need to pass and try to resolve that effect first. You can respond to that triggered ability when you get priority with an instant meant to kill or exile Lurrus.
Lurrus was cast with flash and someone other than the caster is the active player. In this case Lurrus' caster still needs to wait for their turn to cast anything using Lurrus' ability and he can be dealt with. This might work  if it was cast with flash on a non-empty stack during the caster's turn, but not likely as the permanents meant to make use of Lurrus would probably also be under the same effect.
Lurrus' caster is prevented from casting additional spells - there are about a half dozen spells that would prevent additional spells being cast with Lurrus, Rule of Law is the most recent one, Curse of Exhaustion would only affect one player, and Ethersworn Canonist would prevent casting anything other than an artifact after Lurrus, etc.