As written, at 1st level, the VMC’d (as variant multiclassing is generally known) sorcerer gets to... select a sorcerer bloodline. That’s it, they get to pick one. They get absolutely nothing from it at all, it’s just a pointless choice they get to make that will have ramifications later (when they get bloodline powers). Maybe they qualify as having the bloodline for the purposes of prerequisites, but even that is iffy based on how its written and what Paizo has said about it.
To be perfectly honest, I started to write that, then questioned it, read it again, started to delete it, read it again, wrote it back out, then finally gave up and searched to see if I could find anything on the subject, because it’s really poorly written and extremely unclear about what you do or don’t get. This post by Mark Seifter, a Paizo developer, says you don’t get anything that it doesn’t explicitly say you get, and explicitly says that the bloodline arcana is not a thing you get.
You get exactly what it says you get, and not anything else by RAW and RAI. It doesn't say you get bloodline arcana, so you don't get it.
I really don’t like using a random forum post, even from a developer, as a source, but it’s all I’ve got. Mark’s contention that this is “RAW,” as if the rules as written clearly spell this out, is not one I find convincing. Personally, I think the rules as written are horribly vague and I could easily make an argument that the wording of the initial Bloodline entry there actually gives you everything about the sorcerer bloodline, and when the sorcerer gets them rather than delayed as the individual bloodline power entries below it indicate. But the best thing we have here is Mark’s say-so, so that’s what I am going with for this answer.
And ultimately, it does not matter what it does or doesn’t get you, because every single one of the variant multiclassing options is a bad idea 100% of the time. The idea of trading away half your feats for some benefit from another class without having to actually multiclass is sound; I’d even call that pretty good. But the actual benefits of VMCing classes are not worth—not even remotely worth—five feats. Five feats is an incredible amount of investment in anything in this game. It’s literally half your feats, unless you have bonus feats. It is huge. And you could do a lot with those feats.
Some of the VMC options are maybe worth a couple feats. Some aren’t even really that. None of them comes even close to being worth five feats.
So please, I implore you, just... forget they tried this idea. It didn’t work out. It will be bad for your game. Characters that choose not to use it will be better off than characters who do, every single time. There are lots of ways in which multiclassing can cripple a Pathfinder character, but done carefully it can be good. The same cannot be said of variant multiclassing. And as this very question shows, even ignoring the serious balance problems, the rules themselves are just a mess and not well-written.