Is this sentence missing a noun?

I know in some cases a prepositional phrase can be a complement to another prepositional phrase, but this sentence just sounds like it is missing a noun.

So I made myself this little promise that I would reserve my lies for other than my public life. (a line from Primal Fear)

It sounds jarring as it seems to be missing a noun after the preposition "for", for example, "things":

So I made myself this little promise that I would reserve my lies for things other than my public life.

It seems strange to me to have "other than my public life" function as the object/complement to "for". Is a noun necessary here?

It's not missing a noun: it has a pronoun! Here other can be read as being a pronoun.

This is an example of what the OED describes in other sense 7 under section B for nominal (and pronominal) uses:

B. pron. and n.

I. As pronoun.

7. Another thing; something else; anything else. Chiefly in negative contexts, esp. in no other. Frequently with than. Now somewhat archaic or literary.

Here’s one of the more recent citations (which begin all the way back in Bede’s Old English!):

• 1991 Index on Censorship Jan. 6/1
  With their former monopoly virtually intact the old nomenklatura..have little incentive to do other than support present government policy.

Some of the older citations provide some common variants for context:

• 1791 Thomas Paine Rights of Man i. 97
  With respect to the Cour Plénière, it was no other than a medium through which despotism was to pass, without appearing to act directly from itself.
• 1826 James Fenimore Cooper Last of Mohicans I. ix. 121
  With two such examples of courage before him, a man would be ashamed to prove other than a hero.
• 1895 The Westminster Gazette 25 July 4/2
  He thought he could not do other than send the two prisoners for trial.

If those all sound “somewhat literary” to you per their label, then that might be why your original felt funny. But it’s just fine, really.

You can rephrase those citations swapping in anything/nothing but in place of other than:

1. With their former monopoly virtually intact the old nomenklatura..have little incentive to do other than anything but support present government policy.
2. With respect to the Cour Plénière, it was other than nothing but a medium through which despotism was to pass, without appearing to act directly from itself.
3. With two such examples of courage before him, a man would be ashamed to prove other than anything but a hero.
4. He thought he could not do other than anything but send the two prisoners for trial.

Just as one can with your original:

5. So I made myself this little promise that I would reserve my lies for other than anything but my public life.