Why do silver nitrate and sodium hydroxide react to produce silver(I) oxide?
Silver in silver oxide is no more oxidized than in $\ce{AgNO3}$. So you should ask yourself the same question earlier, even before the reaction. Yes, noble metals are somewhat resistant to oxidation. But still they can be oxidized, and thus can form compounds, of which $\ce{AgNO3}$ and $\ce{Ag2O}$ are the examples.

Even gold can be oxidized, though that requires some extreme measures.

As to the formation of oxide instead of hydroxide, there is no clear-cut distinction between Ag and the rest of metals. Many insoluble hydroxides are not quite stable. $\ce{Cu(OH)2}$, for example, would slowly lose water and turn into oxide; likewise, $\ce{Fe(OH)3}$ turns into $\ce{FeO(OH)}$. Silver does the same, just a little faster.