I am wondering what correlation measure to use for calculating the relation between ordinal and scale (interval) variable. For using Spearman's rho the relation must be monotonic. I created a scatterplot, but I am not sure how to interpret it. It shows four vertical lines of similar height. I then created a boxplot, from which I would conclude that it is monotonic, but I am not sure whether it is a correct conclusion? Is there another way to check it and justify that it's monotonic(if it is). I'm not sure if it's relevant here, but the sample is big, N=1500.
For using Spearman's rho the relation must be monotonic.
This is not quite correct. Spearman's $\rho$ is a nonparametric correlation coefficient that assesses monotonicity. It is +1 or -1 if your data are perfectly monotonic. As such, it asks exactly the question you have.
The alternative that comes to mind is Kendall's $\tau$, which is also a rank correlation coefficient.
This earlier question may be helpful in deciding between Spearman and Kendall.
Both Spearman and Kendall have problems with ties, and you have lots of ties. The Wikipedia page for Kendall's $\tau$ has more info on possible ways of dealing with ties than does the page for Spearman's $\rho$. Looking at your data - 1500 data points in just four classes - your findings will quite probably be dominated by how the method you choose deals with ties.
Finally, $N=1500$ is quite a lot. There really doesn't seem to be a lot of structure in your data. If you run an ANOVA and look at $R^2$, I don't think you will be able to explain a lot of variance in your scale variable just from the ordinal variable. However, with a pretty large $N$, both Spearman's and Kendall's correlation coefficient may well be significantly different from zero. Which shows that you shouldn't trust the $p$ value overmuch here. (Of course, there also is the problem about dealing with ties.)