How to open the top cover of a Shimano Tourney ST EF 41?
Page 17 of the dealers manual shows that some of these shifters are not meant to come apart and cables are replaced with an access screw instead of removing a cover.  The manual says not to disassemble further. The shifter may not be able to be easily re-assembled.  That said, disassembling from the bottom would probably be the best way to open the case.  It looks like there are some screws on the underside.

https://si.shimano.com/pdfs/dm/DM-ST0001-05-ENG.pdf

You may be able to remove the screw that gives access to the shifter cable and spray some W-D 40 in there to wash out old lubricant.  Some spray chain lubricant into the same port once the WD-40 dries, might get it working again.  

An internal piece may be broken if the lever is completely immobile.  The pieces inside are intricate and do wear out with use.  my 5600 series front shifter had a pawl break off and it was toast.  no lubing could bring back its function.  the other lever I gave the lube job i described and it has been working smoothly for another year so far.