Looks a right mess, a bad job indeed. The usual procedure for laying sod is, as you mention, soil preparation first, so that would mean digging over, raking off any debris, removing any weed roots, levelling off, then walking all over it on your heels to get out soft spots, re rake/levelling and then laying the turf (sod, we call it turf here...). It is, though, very difficult to get turf added to existing lawn to be at the same height - naturally, after the preparation process, some settlement takes place, so its possible that the new turf will be laid slightly higher than the rest to allow for that. However, it seems your guy made no preparation - what he's done is just lay the turf on top of unprepared soil, and that is never going to match up with any other lawn you have.
I can't see the whole of the lawned area, but if the pre existing lawn isn't large, and if the sods he laid last year are growing healthily, then the easiest thing to do would be to import some topsoil from somewhere, spread it out over the area of the old lawn, till its at the height of the newer turf, then seed it with grass seed, assuming you can get that kind of seed wherever you are. Seed needs loose, friable soil to germinate, to a minimum depth of 2 inches, so if there are areas where the topsoil won't be 2 inches, you'd need to scratch up the surface of the existing lawn to create that.
There are problems with seeded lawns though - they need frequent watering after germination, must be cut with shears the first couple of times (though a hover mower you'll get away with, other mowers rip out the roots) and cannot be walked on or used normally for 3 months. This may not matter if the area is small, but may be an issue if its large - in which case, I'd take the lot up, dig it all over, doing proper preparation, buy more sod/turf and lay it properly... or take up and prepare the pre existing areas and lay new turf there, though then you've got the problem of trying to work out how much settlement will take place to get the whole lawn (new and the turf laid last year) at the same height. There's no quick, easy solution to this one.
UPDATE: Already responded as a comment, but if you're able to, or know someone who can, it should be possible to cut out the new turf, lift and roll it, then prepare the areas beneath properly, and relay the turf at the right level. You'd need to do the job quickly, within a day or two, once the turf has been rolled, unless you have somewhere you can lay it out in the air and keep it watered till you need it. Again with the caveat that settlement might mean it's not level after a few months...
P.S.: If you're going to have a crack at it yourself, but not entirely sure how to do it, buy yourself a copy of The Lawn Expert by D. G. Hessayon - I just checked and its available via Amazon.com around $13... worth every penny (or cent, in your case!)