Can I say that
- changing X from Level 1 (Reference level) to Level 2 will increase Y on an average by 0.406 units
- changing X from Level 1 (Reference level) to Level 3 will increase Y on an average by 0.135 units
If you had no other predictors, yes. (Though your language here is causal which may be misleading if your data didn't come from a design that allows causal inferences. I'd perhaps rather say that the estimated difference in Y between Levels 1 and 2 is 0.406 etc.)
If you had other predictors you can say that the difference between levels 1 and 2 was .406 while adjusting for these other predictors.
The estimates you see in the output are not coefficients in the same sense as a coefficient for a continuous predictor. They are, as you have understood, levels' 2-4 differences from level 1. In a regression with a categorical predictor, you don't get an "overall" estimate for the categorical variable nor an overall estimate for each level. All estimates are comparisons between two levels.
However, the three estimates that you have are also the contrast estimates for level 1, just with reversed sign. E.g. if your reference level was 2, the estimate for level 1 would be -0.406.
(Also as you may notice, your intercept is the estimated mean of Y for level 1)
In a design like yours, it's customary to run post-hoc comparisons (e.g. in R using emmeans package and in SPSS through EMMEANS command). This way you get estimates of all pairwise differences between levels that you are interested in.