I have a First Mate with Warmog's Armor, so he is at 4 Might. I conquer Sunken Temple. Then I get a buff. But can I use the effect of Sunken Temple to draw a card? Because I buffed my First Mate with its conquer effect?

No, you cannot use the effect of Sunken Temple in this scenario.

Here is the breakdown of why:

1. Might Calculation: Your First Mate has a base Might of 3. With Warmog's Armor attached, it gains +1 Might, bringing its total to 4 Might.
2. Mighty Definition: According to the rules (and the text on Sunken Temple), a unit is only considered "Mighty" if it has 5 or more Might. Since your unit has 4 Might, it is not Mighty.
3. Trigger Timing: Sunken Temple's ability is a triggered ability that checks for a Mighty unit at the moment you conquer. Because your unit is not Mighty at the time the conquer occurs, the trigger condition is not met.

Even if you receive a buff after the conquer (from another card's effect), it is too late to satisfy the condition for Sunken Temple. The unit must be Mighty at the exact moment the conquer happens for the ability to trigger.