I have two fields of LU_Name and BETW.
Is there any tool which compares the means of BETW in different groups based on the LU_Name nominal field and reports their significant correlation?
As BERA suggested, Summary Statistics will do the trick for the means per Group. Alternatively, you can right click on the BETW field and choose "Summarize". In the window that pops up select BETW in the first option and then, on the second option, locate and expand LU_Name and select "Average". It will yield the same output as Summary Statistics, it's just a slightly faster way to do it.
Furthermore, for the correlations, I am not aware of a tool in ArcMap that does this directly. What you can try is use Table to Excel to export the attribute table into an Excel file and then easily calculate the correlation between two columns (in case you use Libre Calc, it is just as easy). You can then decide the significance of the correlation based on your sample size and the significance level.