DIY here usually means actually making your own juices
You do not need a syringe to fill a cart
By cart - do you mean cart aka cartridge which pushes onto or into an atomizer?
Or cartomizer aka carto which has its own internal atomizer coil, holds juices and screws right on the battery?
If a cartomizer - does it have stuffing or not? Those without stuffing do need a syringe.
What model? And what's the PG/VG of your
juice?
With carts, you just drip into the stuffing of the
juice cup section of the cart until the top looks damp but not a puddle. Touch tissue to it if it ends up being a puddle. If you over fill, the cart will be more likely to suddenly dump its juice. NOTE! If you have an ecig with carts that go OVER the atomizer, then your juice cup may be recessed. In that case hold the cartridge (and this only for cartridges, not cartomizers) and looking in the open end, stick an unbent paperclip in the mouth hole and push up the recessed juice cup. When done filling make sure the stuffing is even with the top of the juice cup and then get the juice cup back down into the mouthpiece without pressing on the stuffing. Then put it on the atomizer.
With stuffing-type cartomizers...
Carefully remove the end cap (I stick a paperclip leg into the mouth hole for 2/3 to 3/4 the depth of the end cap, and pry out the end cap without catching any stuffing or wires underneath)
When I fill the first time (and whenever adding a lot of juice) I hold tissue against the threaded end hole of the carto. Tilt and drip in 18 drops slowly (wait after the first few which may take longer to soak in). Then I wait 15-20 seconds. Then I see how many more drops the stuffing will soak in easily. Then I do cleanup of the threads, then tilt open end down into a tissue and puff into the threaded end hole to make sure the air channel is clear, then while still pointed down I touch soft tissue to the top of the stuffing to sop up any excess (helps prevent leaks). Then to settle it I let it lay on different sides for 15-20 minutes to spread the juice to every area of the carto. 15-20 minutes is more than enough settling with the thin juices I use. Only the first fill needs some settling time (to make sure every bit of stuffing is damp).
As I vape, I keep it damp. If the vapor seems light or dry or tastes even a tiny bit off or if the carto is getting abnormally warm, I see if it could use some juice. And you can also check the top of the stuffing - if it has gone white you could probably add a bit of juice before heading out or putting a new battery on it etc.
And in stuffing cartomizers I use thin clear juices, PG w/ 20% or less VG, and I avoid juices that leave a lot of flavoring sediment in the stuffing.
And I recommend never putting an untried juice in a carto. Only put juices you know you like into cartos.