In this scenario, we're dealing with an isochoric process, meaning the volume of the gas remains constant. For an ideal gas in an isochoric process, the pressure ($P$) is directly proportional to the temperature ($T$) in Kelvin.

The relationship can be expressed as:

$ P \propto T $

Given the percentage increase in pressure is $0.4\%$ when the temperature is increased by $1^{\circ} \text{C}$, we can use the relationship:

$ \frac{\Delta P}{P} = \frac{\Delta T}{T} $

Substituting the given values into the equation, we have:

$ \frac{0.4}{100} = \frac{1}{T} $

Solving for $T$, we find:

$ T = 250 \, \text{K} $

Therefore, the initial temperature of the gas in the vessel must be $250 \, \text{K}$.