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It seems to add -ment suffix does not generally change the spelling of the base word: "mange" -> "management".
However, I've just noticed that this is not the case for "argue"->"argument". I wonder if there is any other instance similar to this and is there any rule regarding the spelling of suffixes starting with a consonant.
This seems to have been a variation of spelling that developed in Middle English.
The word derives from Old French, the Old French Spelling was "arguement", with an "e". That was derived from the Latin "argumentum", (argu- (make clear) + -mentum). Note the Latin doesn't have an "e"
The French "e" came to be inserted as the related verb ending is -ere in Latin and -er in French, (as "arguer") French tends to insert an "e" when forming "-ment" nouns from "-er" verbs. The "e" was then dropped in English, probably to better match the original Latin.
Words like "management", for example, are not based on any Latin original. They were formed in English from "manage"+"-ment", and haven't lost any letters.
The only rule is that there is no rule, but Latin can sometimes justify how a word came to be spelled.
I assume you're asking why the e is removed when the suffix -ment is appended. There are no fixed rules that I'm aware of, but here's a general rule of thumb:
The suffix -ment and -able don't often change the pronunciation of the root word.
The e from manage doesn't get removed because it will chang the pronunciation of manage:
the pronunciation doesn't change because the e is there to make the g soft (/d͡ʒ/)
But if you remove the e:
it will change the pronunciation of the word i.e. g usually represents a hard g sound (/g/) before a, o, u and another consonant.
In argue, the final e is there because English words don't often end in a u (or v).
In argument, there are other consonants after the u so we don't need the e:
Another example is:
you can't remove the e because if you removed it, you'd get noticable with a hard c sound (/k/)
Updatable and updateable both are correct because the removal of the e doesn't change the pronunciation of the root word.
In summary, if the removal of the e changes the pronunciation, keep it, otherwise remove it.
