(Question was edited to reflect: Why is a question on Inkjet refill Off Topic for Superuser?)
I've done this before. Typically refilling syringes are huge, and have the pointy bit of the needle reversed so its flat tipped and you can't stab yourself with it.. My local ink dealer tends to supply both squeeze bottles and syringes on request. 'Single' use syringes are just plastic, and should be fine, though true single use syringes for medical use are just coming out on the market.
Typically there's a re-filling hole under the label somewhere. Remove the label (carefully) and probe to see where the needle can go in all the way. Withdraw the needle to check - unless its bone dry, you will see some ink.
Reinsert the needle Saturate the sponge until ink just starts to flow out, then replace the label or tape over the hole if it was covered. Don't bother flushing out the old ink. There's no real point. I've never done it over the years I've refilled ink on my printers.
As for whether its the right ink? It depends. My dealer just sells it by brand.
At the end of the day though, get some ink, see if you're happy with it, and if you are, just stick to the same maker. Unless you happen to have a gas chromatograph...