I start with some general information and answer your questions below. I'll assume continuous compounding.
The bond price is given by
\begin{align*}
P_B=\sum_{i=1}^n c_{t_i} \cdot e^{-y\cdot t_i},
\end{align*}
where $c_{t_i}$ denote the $n$ coupon payments occurring at time points $t_i$ and $y$ the yield-to-maturity. Note that the last payment $c_{t_n}$ included both, the final coupon and the bond's face value.
The (Macaulay) duration is given by
\begin{align*}
D &= -\frac{1}{P_B} \frac{\partial P_B}{\partial y} \\
&= \frac{1}{P_B} \sum_{i=1}^n t_i\cdot c_{t_i}e^{-y\cdot t_i}.
\end{align*}
The duration may be interpreted as weighted average of the time points $t_i$ when the coupon payments occur with unit ''year'' and ''weights'' $\frac{c_{t_i} e^{-y\cdot t_i}}{P_B}$. Note that these weights add up to one. Alternatively, the duration approximates the change in the bond price given a change in the interest rate, i.e. $\Delta P_B\approx -D P_B \Delta y$. Using discrete compounding, you'll need the modified duration here. For larger changes in the interest rate, you may want to include the bond's convexity.
As a special case, consider a zero-coupon bond with $c_{t_i}=0$ for $i<n$ and $c_{t_n}=1$. Thus, $P_B=e^{-y t_n}$ and $D=t_n$. As there are no payments, the weighted averages of the coupon payment dates is simply the bond's maturity: that is how long you have to wait until you receive cash flows. Note that the bond price can be solved for the bond's yield explicitely, i.e. $y=-\frac{1}{t_n}\ln(P_B)$. This is in general not possible for coupon bonds whose yield-to-maturity is typically found numerically.
Now, let's talk about your questions...
- The bond price $P_B$ is monotonically decreasing in the yield $y$. The higher $y$, the lower the present value of the individual coupon payments, $c_{t_i}e^{-y t_i}$. Hence, we observe a negative/inverse relationship between bond price and bond yield.
- The duration of a zero-coupon bond is simply the bond's maturity. Hence, the shorter the time to maturity, the lower the bond's duration. This makes sense. A zero-coupon bond which matures quite soonish is hardly sensitive towards interest rate changes. A coupon-bearing bond has an even lower duration than a zero-coupon bond since you already receive some cash flows (coupon payments) prior to the bond's maturity.
