Copying your PSP save data
Plug your PSP into your PC with a Mini USB cable (the kind that fits into the top of your PSP) and select the USB Connection option in the PSP System menu.
The PSP should show up as a connected drive in your My Computer menu. It's probably called "Removable Disk" if you're on Windows. Open that drive and navigate to the PSP folder.
Copy the SAVEDATA folder onto your PC somewhere like you Desktop. That folder has all your PSP saves and we'll copy them to the PS Vita soon. If you're concerned about how much space the data will take up on your PS Vita, check how large this SAVEDATA folder is, you're about to copy that much data onto your Vita. If it's too large, we can only copy the specific games you want the data for later.
Prepare your PS Vita to copy the files
Download any one PSP game onto your Vita and play it enough to generate a save file. Any game will do, it doesn't have to be a game you want to copy the save data for.
Install the PS Vita Content Manager Assistant for Windows or Mac OSX.
Connect your Vita to your PC with the USB cable and open up the content manager. Pick Copy Content: PS Vita System -> PC -> Applications -> Save Data (PSP/Other).
You should see the game you played earlier in a list; tap the PSP game to select it and hit copy. Click Okay to copy it onto your PC.
Find the folder on your PC the Vita copied to; it should be in your Documents folder in a folder named PS Vita, in that folder there's a folder called PSAVEDATA. In that should be a single folder named with random letters and numbers. Open that folder; this is where we're going to copy the PSP Save Data.
Copy the files onto your Vita
Open up the SAVEDATA folder from earlier, those are your PSP saves. Also open the PS Vita folder we mentioned before that has your PS Vita's copy of the save files (you want to be inside the folder with the name that's random letters/numbers). Copy and past ALL folders from the SAVEDATA folder into the folder named with random letters. Say Yes to any dialogs that ask if you want to merge/copy files; this means you're overwriting the Vita's copy of the save file with the PSP one.
Now just open Content Manager on the Vita again and pick Copy Content: PC -> PS Vita System. Pick Applications -> Saved Data again. You should now see all of your PSP save data. I recommend hitting Select All and just copying all of this data over. You can also pick and choose just the games' data you want to copy over if you want to save yourself some space.
Pick Copy and hit Okay again, it will ask you if you want to overwrite any data you have on the PS Vita for the PSP games; you want to do this, so click Apply to All and tap Overwrite.
Your PS Vita now has all save data from your PSP and you can keep it there forever, and it's also stored with your PS Vita backup. Any time you download a PSP game onto that Vita, it will start with the save data you copied from your old PSP, no need to repeat this for each game!
Best Answer
The vita has a built-in psp emulator. The psp games you can play you have to buy from psn first. Some games offer dlc, but for some it doesn't work, or is not available on psn. It's all a bit sketchy. I you just want the dlc, I suggest investigating network issues to make your psp work first, if you are using WPA security, try switching to WPA2 because WPA is obsolete. If this doesn't work out for you, do some googling to see if others were able to get the dlc working on their vita.
In short: It is most certainly possible to get the game and the dlc on your vita, but it all depends on Sony if it will work or is even available.