I am travelling from Korea to India with one self-transfer stop at Vietnam and just found out I needed a Viet e-visa for this. I applied for an emergency e-visa and it says it will be available 1 hour after my boarding time. Will I be okay? This flight cost me way too much of my meagre salary and I can't afford to book again.
Presumably by a "self-transfer stop" you mean that you are travelling on two tickets? In general most nationalities do not require a visa to transit Vietnam (presuming you are going via either Ho Chi Minh City or Hanoi), however with multiple tickets, and especially if you have checked bags, then you may well need a visa.
If you require a visa, then you will generally need to have it before you will be able to check-in for the first flight in your trip, or at the latest before you board that flight. (The exact process varies from airline to airline). There is zero chance that the airline will allow you to board before the visa is approved if one is required.
Unfortunately if your visa isn't approved earlier than stated then your only option is likely to change flights to a later date.