First, you should have U100 (not U300) syringes. Lantus is a U100 insulin -- the concentration of the insulin and the syringe calibration need to match.
Most people do get good at eyeballing the quarter unit amounts. However, many people use digital calipers to help measure a dose consistently. You can purchase 6" digital calipers from Harbor Freight. (I'm sure there are other places, too.)
@Marje and Gracie put together information on
how to measure with calipers.
I'd encourage you to read the sticky notes at the top of the Lantus forum, if you've not done so already. There are photos of what the syringe looks like with quarter unit doses so. you have a comparison. The stickies will also help to reduce at least some of your anxiety. With Lantus, the basic mantra is "shoot low to stay low." Because it is a less harsh insulin than Vetsulin, you have a great deal more flexibility. Onset is 2 hours after you shoot -- you do not need to feed your cat 20 - 30 min. before shot time. Lantus typically doesn't drop numbers like a rock -- it can happen but it's far more typical of a shorter acting insulin.
There are also a lot of people who are members of the Lantus forum. They are very generous with their information and are there to help.