Why isn't the center of the galaxy "younger" than the outer parts?
The gravitational potential of the disk of the Milky Way can be approximated as:
$$ \Phi = -\frac{GM}{\sqrt{r^2 + (a + \sqrt{b^2 + z^2})^2}} \tag{1} $$
where $r$ is the radial distance and $z$ is the height above the disk. I got this equation from this paper, and they give $a$ = 6.5 kpc and $b$ = 0.26 kpc.
In the weak field approximation the time dilation is related to the gravitational potential by:
$$ \frac{\Delta t_r}{\Delta t_\infty} = \sqrt{1 - \frac{2\Delta\Phi}{c^2}} \tag{2} $$
At the centre of the galaxy $r = z = 0$ and equation (1) simplified to:
$$ \Phi = -\frac{GM}{a + b} \tag{3} $$
No-one really knows the mass of the Milky Way because we don't know how much dark matter it contains, but lets guesstimate it at $10^{12}$ Solar masses. With this value for $M$ and using $a$ + $b$ = 6.76 kpc equation (3) gives us:
$$ \Phi = 6.4 \times 10^{11} \text{J/kg} $$
Feeding this into equation (2) gives:
$$ \frac{\Delta t_r}{\Delta t_\infty} = 0.999993 $$
So over the 13.7 billion year age of the universe the centre of the Milky Way will have aged about 100,000 years less than the outskirts.
The centre of the galaxy will indeed appear to pass through time more slowly than the edges, but the effect will not be great.
Because the Einstein field equations are very difficult to solve, it is not possible to calculate the exact magnitude of the time dilation, but we can make an approximation. By assuming that the black hole at the centre of the galaxy is electrically neutral and non-rotating, and ignoring the effects of all other mass/energy, we can calculate the time dilation at a distance $r$ from the galactic centre, as seen by an observer at infinity.
The formula for this time dilation is $\Delta t_0 = \Delta t_\infty \sqrt{1 - \frac{r_S}{r}}$, where $t_0$ is the proper time at a distance of $r$ from the galactic centre; $t_\infty$ is the proper time measured at infinity, and $r_S$ is the Schwarzschild radius of the black hole that lives at the centre of the galaxy. Because $r_S$ is many times smaller than $r$ (except for any unlucky stars finding themselves being eaten by the black hole), we would not see any appreciable difference in the rate at which time passes between stars close to the centre and those far away.
All of this analysis assumes that Sagittarius A* is exactly at the centre of the Milky Way, which is not exactly true. The distance between the two will cause the actual centre to be slowed by the gravity of the black hole, just like anything else. This will be highly dependant on the proper distance between the centre and the hole, but could be calculated - with some approximation - by the above formula.