How old is the phrase "What's your angle?"
Have you checked Google Ngrams? That might help.
The Ngram indicates the phrase what's your angle first appeared in the mid-1920's, so you should be okay. However, when using this tool for such verification purposes, it's important to remember that Ngram results are based off of written documents (magazines, journals, and books); there's a chance the expression may have been used conversationally before it made its way into print. Some sayings need to reach a critical mass of usage before writers start using the phase in their work.
Old newspapers
Searching the Chronicling America newspaper archives (1836-1922) gave one result. This question was asked in a letter to the New-York Tribune of December 20, 1922:
Dear Sir: A day or so ago we had an argument as to what the five leading features for the year were in sport, whether on the part of an individual or a team. We never seemed to get anywhere with the debate. What's your angle on the top five? F.L.G.
Oxford English Dictionary
Whilst not the exact phrase, the OED's sense 9b is:
The perspective or aspect from which something is considered, regarded, or presented; a viewpoint, a standpoint. Freq. with preceding modifying word.
With pre-1930 quotations:
860 E. G. Parker Reminisc. Rufus Choate viii. 480 From whatever angle he looked at the facts, from whatever chords he struck the tones, you heard ever the same recurring strain.
1872 ‘G. Eliot’ Middlemarch II. iii. xxiii. 13 Tacit expectations of what would be done for him by Uncle Featherstone determined the angle at which most people viewed Fred Vincy in Middlemarch.
1907 E. Wharton Let. 19 Nov. (1988) 124 As soon as I look at a subject from the novel-angle I see it in its relation to a larger whole, in all its remotest connotations.
And 9c:
colloq. (orig. U.S.). A way of approaching a task; a modus operandi, a method or scheme (sometimes with implication of dishonesty or exploitation).
1920 J. Conway in Variety 31 Dec. 8 He finally caught the proper angle... When in Rome do as the Romans do.
1921 J. Conway in Variety 18 Mar. 5, I thought I was hep to all the angles.