Ensembl BioMart is probably your best bet.

You may need to use/write a quick script or awk command to extract your chromosome number and position but you can enter a list of chromosomal positions in the filter menu under regions->multiple chromosomal positions. You can then choose which attributes you would like to download such as Ensembl Gene ID or (under the external references menu) Refseq id.

If your list is too long you'll probably have to use the Perl API. Short tutorial available here: http://www.ensembl.org/info/data/biomart/biomart_perl_api.html

You need the variant effect predictor, does exactly what you need.