Is "alas on me" correct?

alas

interj.

Used to express sorrow, regret, grief, compassion, or apprehension of danger or evil.

[Middle English, from Old French a las, helas, ah (I am) miserable, from Latin lassus, weary

American Heritage Dictionary

The use from Old French seems to have some impact on the English use of alas as it is able to take a content clause:

Alas that you should be insulted with dried-apple pie and molasses preserves! Oh, horror! I thought that you would have fresh fruit and vegetables. (Atlantic Monthly: May 1896: 649-660)

His empty flat resounded silence . Alas that he had ever come to this house. (The Tenants, Bernard Malamud,1971)

Alas that it should have been possible to say, even with rhetorical exaggeration, that when he went to Europe he was the greatest man that ever lived, and when he returned he did not have a friend.(Brains win and lose: Woodrow Wilson, Gamaliel Bradford, The Atlantic Monthly: Feb 1931: p. 152)

This use of alas is very similar to curse or damn when used in expressing our feelings about some situation.

It would not follow, however, to use alas as we might say a curse on you. Here we are not passing judgement or, as per the definition above, expressing sorrow regret, grief, compassion or apprehension of danger or evil but instead making an appeal or prayer for evil or misfortune to befall someone or something (AHD).

In short, an alas would be difficult to understand, especially in the context of an alas on me as it might be interpreted as requesting others to express sorrow, regret, grief, compassion, etc. for you, and not the curse that you wished to call upon yourself.

Alas is an interjection that is usually used just on its own, without a preposition, to express sadness or regret. In this case, it means something like "Oh, no!" or "oh, dear" or "how unfortunate!", and it is set off alone by commas, or even used as a single-word sentence: Alas, James had finals and couldn't attend. (Grammarist). Alas! What are we to do now? (Cambridge).

It can also be used as an adverb to mean "sadly" or "unfortunately" (Lexico). Alas, it's not that simple. (Collins).

Woe is me is an expression that might better suit your needs. It is used to express how unhappy you are, yourself. I'm cold and wet and don't have enough money for the bus home. Oh woe is me! (Cambridge). It's often listed, in fact, as a synonym of alas (Thesaurus).

That said (and as you point out in your question), you have to be careful with archaic poetic terms. They can come across as overdramatic, humorous, or ironic. Regarding woe is me, several dictionaries point this out right upfront. MW: "used in a humorous way to say that one is sad or upset about something". Dictionary.com: "Woe is me is an over-dramatic, often comical way to express sadness or disappointment at an unfair situation." Cambridge qualifies the expression as "old use or humorous".

If those connotations don't interfere with your purpose, woe is me might work for you. While it doesn't use a preposition like "on", it does use a form of the first-person singular pronoun, me, and allows you to associate the sentiment of regret clearly with yourself.

If you don't care about preserving the tone of alas and want only to convey the sense of a curse upon yourself, you might say a pox on me. Because, like alas, it's an old expression, it can also be seen as overdramatic or ironic. But it can be used with a preposition and the pronoun me, and it clearly expresses a wish that something bad happen to someone (Wiktionary).