From the beginning of chapter 10 of The Just Men of Cordova (1917) by Edgar Wallace, the narrator was describing someone, saying:
But he certainly lived like a gentleman, as all Somers Town agreed, for he went to bed at whatsoever hour he chose, arose with such larks as were abroad at the moment, or stayed in bed reading his favourite journal.
I found that "rise with the lark" means " wake up with sunrise", but I can't get the whole meaning of this bolded statement.
It is straightforward: it is taking the expression He rose with the larks literally - getting up when the skylarks take to the sky - and considering that maybe not all the larks were in the air at the same time, so limiting it to such larks as were abroad at the moment i.e. "those larks that were about at the moment".
So it uses the literary syntax such ... as ... meaning "those ... which ...".
And it is playing literary games with the familiar expression, noticing that when taken literally it implies that all larks take off at the same time, so modifying the expression to be more precise: as I say, playing a game.
It's saying he gets out of bed whenever he wants to. Instead of (metaphorically) first finding out when the larks rise and then getting out of bed at that time, he first gets out of bed whenever he wants and then (metaphorically) observes if there are any larks.
Note that "rise" means getting out of bed and starting the active part of the day, not just waking up.
You can tell this (a little) from context, because he "went to bed at whatsoever hour he chose" (if you go to bed at different hours then you won't wake up at the same time every day) and "arose with such larks as were abroad at the moment, or stayed in bed reading his favourite journal" (sometimes he wakes up and gets straight out of bed, sometimes he wakes up but stays in bed reading).
Also "lived like a gentleman"--in 1917 a stereotypical "gentleman" was a member of the upper class, i.e. someone who was well-off financially, who didn't need to work a normal job for a living, so a "gentleman" had the luxury of choosing when to get out of bed instead of working on someone else's schedule. Actual individual "gentlemen" could be different, but as a class it means well-off. Today we might say "lived like a trust fund baby" or "lived like a movie star".
[...] he went to bed at whatsoever hour he chose,
arose with such larks as were abroad at the moment, [...]
The parallel construction here is your clue into the joke. The gentleman in question goes to bed when he pleases and gets up when he pleases. The narrator compliments the gentleman for "arising with the larks" (a phrase that, you're right, normally means "arises early in the morning")... but his next words indicate that we don't mean the usual early-morning larks. We mean, you know, whatever larks might happen to be out at this hour.
A common joke with the same format is, "I start drinking only at 5 o'clock... and it's got to be 5 o'clock somewhere!" (However, in that case, the joke is that it's 5 o'clock somewhere else in the world, perhaps far away, which coincidentally is another meaning of the word "abroad." In the context of this quotation, the word "abroad" simply means "out and about, conducting their daily business.")
The narrator then goes on to say (and IMHO weaken the punchline of the joke by saying) that in addition to getting up with whatever larks are abroad at the moment, sometimes the gentleman stays in bed instead. So:
He gets up with the lark, but it's not always the morning lark. And, come to think of it, sometimes he doesn't get up at all.
I find this leaning towards the lines of "woke up with such joy as if he were on a vacation or rather care-free and ready to take on the day," which is in alignment with its contradiction in the next statement which says, "or stayed in bed reading his favourite journal."
