How to use Lewis theory to differentiate between an acid and a base?
First of all, within the Lewis framework, you may have substances that can act as both acid and base (though most of the time as one or the other, not both at the same time).$^{1}$ The adjective for such substances is "amphoteric".

It comes down to a bit of chemical intuition to decide which is which. It is true that almost anything that has a lone pair can act as a base - though it may take a strong acid to form an actual Lewis pair. For acids, one should look for an obvious gap in the electron cloud ($\ce{AlCl3}$) or candidates for complex formation ($\ce{FeCl3}$).$^2$

For nitric acid, the situation is pretty clear: the hindrance provided by three oxygen atoms around the central nitrogen prevents access, so it will hardly act as a Lewis acid. It can act as a base, however, weak Brønsted bases are usually also weak Lewis bases because they are content on their own.

For sodium hydroxide, one must realize that in an aqueous medium, the two ions will fully dissociate. The hydroxide anion will act as a Lewis base, but the sodium cation with its full outer electron shell will hardly act as an acid.$^3$



$^1$ Adducts of e.g. $\ce{CO2}$ in frustrated Lewis pairs, wherein the $\ce{CO2}$ acts as a sort-of bridge between the two Lewis sites, are an example of when "both at the same time" can occur.

$^2$ I am aware that these compounds will form dimers or more complex structures in reality. If you are able to spot this behavior and understand why, you are already able to spot Lewis acids.

$^3$ Note that the sodium cation in water will be solvated, i.e. some water molecules will orient their lone pairs towards the ion. The border between solvation and Lewis adduct formation can be blurry.