Outrunning a light-speed signal for a limited amount of time by accelerating and then coasting

The problem doesn't have a solution with the given parameters. The reason is that acceleration at $5g$ has a curvature radius in spacetime of $c^2/5g approx 71text{ days}$, which is larger than the 60 day separation between the light and the ship. So the light will reach the ship while it's accelerating, in a lot less than 80 years.

If you increase the acceleration or the head start a bit then the problem has a solution, though it will require accelerating to an implausibly high speed to outrun the light for 80 years.

Say the light's path is $x=ct$, and the ship starts at rest at $x(0)=x_0,,t(0)=0$, accelerates at $a$ for a proper time $τ_1$, then coasts.

After the acceleration, the ship's rapidity is $α = aτ_1/c$, its position is $x(τ_1) = x_0 + (c^2/a)(cosh α - 1)$, $t(τ_1) = (c/a)sinh α$, and its velocity is $x'(τ) = c sinh α$, $t'(τ) = cosh α$. Its position at any later time is therefore

$$begin{eqnarray} x(τ) &=& x_0 + (c^2/a)(cosh α - 1) + c(τ-τ_1)sinh α t(τ) &=& (c/a)sinh α + (τ-τ_1)cosh α end{eqnarray}$$

Setting $x(τ) = c t(τ)$ and some manipulation gives

$$x_0 + (c^2/a)(e^{-aτ_1/c} - 1) = c(τ-τ_1)e^{-aτ_1/c}$$

or (solving for $τ_1$)

$$τ_1 = τ - frac{c}{a} left( 1 + W(e^{a τ/c - 1} (a x_0/c^2 - 1)) right)$$

(where $W$ is the Lambert W function).

As an example, with $x_0 = 60text{ days}$, $a = 6g$, $τ = 80text{ years}$, I get $τ_1 approx 603text{ days}$. In that time, they'll accelerate to a rapidity of about $10.2$, which is about $0.999999997c$.