Hi! You can create a random points layer and extract values from both rasters. Next, you can use any statistical software to calculate the Pearson r between two datasets.
Yes. You definitely can. I'm currently working on a similar research for a study area. On your ArcGIS, create a shapefile for the representative observational points. Afterwards search for "extract multiple values to points" in your toolbox. This tool will enable you pick the LULC and LST rasters you are interested in, and extract the values from the rasters to those points. The outcome is an attribute table which shows the LULC data and corresponding LST values for each observational point.