A possible analytical approach.
1. If we want to describe the behaviour of $y(t)$ as $t \rightarrow \infty$ then we need to actually find out what $y(t)$ is. We can do this through the method of integrating factors.
Given $y'(t) - y(t) = e^{-t}$, let $\mu(t)$ represent the integrating factor. $\mu(t) = e^{\int a(t)dt}$, where $a(t) = -1$. Therefore,
$$ \mu(t) = e^{\int -1dt}, \qquad \mu(t) = e^{-t}, \qquad y'(t) - y(t) = e^{-t}$$
$e^{-t}(y'(t) - y(t)) = e^{-2t}$
$\frac{d}{dt}(e^{-t}y(t)) = e^{-2t}$
$\int \frac{d}{dt}(e^{-t}y(t))dt = \int e^{-2t}dt$
$e^{-t}y(t) + c_1 = -\frac{1}{2}e^{-2t} + c_2$
$e^{-t}y(t) = -\frac{1}{2}e^{-2t} + k$
$$y(t) = \frac{-\frac{1}{2}e^{-2t} + k}{e^{-t}} = -\frac{1}{2}e^{-t} + k e^t$$
Now that we have an equation for $y(t)$, we can describe its behaviour.
2. Zhihong I think this is what you were talking about where we could find the limit:
$$
\lim_{t \rightarrow \infty} \frac{-\frac{1}{2}e^{-2t} + k}{e^{-t}}
$$
which will depend on what we select $k$ to be. You can then see for what values of $k$ how the function $y(t)$ behaves and this should of course agree with the direction field but be more analytical and accurate.
