"Get to sector 5 without upgrading your reactor in the Zoltan Cruiser."
To earn this achievement, the ship's reactor cannot be upgraded at all through the ship menu -> upgrades tab -> Reactor section before it reaches the fifth sector. Events which feature offers to upgrade your reactor, advertised or not, do not count against this achievement. Systems and subsystems can be upgraded, but only Zoltan crew members, backup batteries and random events that upgrade the reactor can provide more power without invalidating a run.
Purchasing Cloaking would help your survival during early-game, as Leto Missiles combining with Halberd Beam would be sufficient for defeating most easier opponents. Power required to run the Cloaking System could be acquired by either powering down non-critical Systems like Medbay or Shield, or by hiring/rescuing Zoltan crews.
Someone trying to unlock Manpower can make a "pacifism run" and attempt to earn many of the "Going the Distance" achievements at the same time by just fleeing every battle as quickly as the FTL drive can charge. However, this would make it fairly difficult to beat the game in the same run.
Trying to get this on FTL: Advanced Edition makes it much easier getting the achievement. Getting a backup battery allows having more energy for a short time and will not lock you out of this achievement. The Advanced Edition also features events that can extend your generator capacity which will not prevent you from getting the achievement, like at a beacon where a ship requests fuel.