Discretized Time?
If we take $t=0$ and $t+\delta t=\tau$, you have payouts of:
$$
\begin{align}
\text{Payoff 1} &= \omega_1 \left(K_1 - \frac{100}{S_\tau}\right) \quad \text{and} \\
\text{Payoff 2} &= \omega_2 (S_\tau - K_2).
\end{align}
$$
Payoff 1 clearly increases with $S_\tau$ as does Payoff 2. (Imagine the underlier at 100 and then it goes up to 101: Payoff 1 would be $\frac{100}{100} - \frac{100}{101}>0$.) Therefore, to hedge Payoff 1 you would sell some of the contract offering Payoff 2.
Continuous Time?
However... I suspect the $t$ is, as you say, a time index which continues to increase. Then, Payoff 2 is clearly a "delta-1" investment. However, your discretized statement of Payoff 1 is very odd. So, expressing everything in continuous time (no more $t+\Delta t$), I suspect you mean that
$$
\text{Payoff 1} = \omega_1 \frac{-100}{S_t}.
$$
In that case, we can look at the derivative of the intrinsic value:
$$
\frac{\partial \text{Payoff 1}}{\partial S_t} = \omega_1\frac{100}{S_t^2}.
$$
In this case as well, you would sell some of the contract offering Payoff 2 to hedge holding the contract offering Payoff 1.
Log-Payout?
Finally, there is a slight chance this discretization is coming from considering a contract paying off $\log(\text{underlier})$, since the derivative of $\log(S_t)$ is $1/S_t$). If that is the case and your discretized payoffs were just a sort of Taylor Series approximation, then Payoff 1 will be related to the volatility of $S-t$.
