Wouldn't a negative mass be going faster than $c$ according to our current models of relativity?

It is difficult to know exactly what “moving backwards in time” means so I am going to interpret your question as “does replacing mass with negative mass give the same result as replacing $t$ with $-t$ ?”.

Newton’s second law $F=ma$ does not change if you replace $t$ with $-t$. In other words, if a force accelerates a mass from $v_1$ to $v_2$ then if we reverse time the same force accelerates the same mass from $-v_2$ to $-v_1$ - the acceleration is the same, and is still in the same direction as the direction of the applied force.

But if we replace $m$ with $-m$ then Newton’s second law becomes $F=-ma$. In other words a negative mass will accelerate in the opposite direction to the applied force. This is a different outcome than just reversing time.

In relativistic quantum mechanics it can be shown that time reversal operator commutes the same way as parity inversion operator :

$$ {\text{T}}H{\text{T}}^{-1} \equiv {\text{P}}H{\text{P}}^{-1}$$

Where $H$ is energy operator, $T$ - time reversal operator and $P$ is parity transformation operator. What this means ? Consider this picture of a cannon shooting a projectile :

In A pic a time is reversed and in B pic - $x$ coordinate is reversed. It appears that both these cases produce same transformation to physical laws at hand. Or in layman terms, if you watch a projectile flying backwards in a movie, you can't distinguish if it is due to a movie being played backwards or simply due to a movie frame image flipping along X axis. Both transformations produce same effect.

In the quantum mechanics there can't be negative energy, because lowest possible energy is that of vacuum. It is greater than zero and can't be lower than that.

But if we look at it from classical Physics eyes, just out of curiosity, then according to second Newton law change in a particle speed can be expressed as :

$$ \Delta v = m^{-1}F~(t_2-t_1) $$

So if you want to reverse a particle speed, keeping force fixed, you have to either reverse a time flow ($t_2 < t_1$) OR put a negative mass $-m$ into equation.

However it should be noted that negative mass is highly speculative thing. Because it is shown that two $+$ and $-$ masses would produce a "Runaway motion". Positive mass would be repelled from a negative one, but a negative one would be attracted to positive one at the same time ! This would put system in constant "runaway", where negative mass indefinitely tries to catch a positive one. Thus producing system self-acceleration with no external force or field applied. Put these opposite sign masses on a wheel, and you will get a perpetual motion machine. Which will break many laws, such as first or second law of thermodynamics or even a general relativity, because rotating device will get more massive for no apparent reason. (Albeit particle system total momentum and total kinetic energy remains zero). Gas composed from $+-$ mass particle mixture would also act very strange,- gas part composed of positive mass would increase in temperature without a bounds as well as second part composed of negative mass particles would gain a negative temperature also without a bound, balancing each other. There are some attempts to solve this runaway motion paradox, however this is still "a work in progress".