If you use the right type of microarray, yes.
For example, the human exon array is one such option: http://www.affymetrix.com/estore/browse/products.jsp?productId=131452&navMode=34000&navAction=jump&aId=productsNav#1_1
You should always be able to check the array design to see if the probes target a specific transcript. But you can't (and shouldn't) generally assume that your typical gene expression array can detect alternative splicing.
Alternative splicing is certainly theoretically possible with RNA-dat, although I would recommend focusing on specific splicing events (like exon-skipping, intron retention, etc.). Paired-end RNA-Seq data is probably a good idea, but single-end might be OK if you have long enough reads (at least >100 bp). My favorite tool that I have found so far is MATS:
I haven't been very satisfied when I found potentially interesting differences between transcripts for different genes - when I visualized the alignments in IGV, it looked like the differences in estimates came from low coverage regions of the gene that I didn't think really justified a clear difference between transcript expression.
A great place to check for this is on ensembl, they have many microarray probe sets annotated for the specific transcripts they target. You should be able to work backwards from your variant of interest to see if there are any available microarrays.