I have used Kaplan-Meier method several times before when I compared how group $A$ survived compared to group $B$ through a period of (say) 5 years.
Now I face a somewhat different scenario: I have some score variable $v$ which is continuous, and I want to know how it affects survival?. Since I know $0 < v < 100$ I simply categorized it to ten groups and did a Kaplan-Meier on them.
I wonder if there's a better way? maybe Cox Regression?
Your sense that Cox regression is a better solution is correct.
It's generally not a good idea to break up a continuous predictor variable. One useful approach is to use a flexible form like a spline to model the continuous predictor. That lets you discover possible non-linear relations between the predictor and outcome without using up too many degrees of freedom. Your 10 groups use up 9 degrees of freedom. In contrast, a continuous spline fit with 4 or 5 knots, usually sufficient to capture nonlinearities well, would use up less than half as many.
You can use the spline fit to display the modeled continuous relation of outcome to your predictor. If your audience wants to see full survival curves, then you can illustrate with groups separated by values of the predictor. But that grouping should be limited to display; statistical analysis should be done on the model based on the continuous predictor.