in able to do a remote shutdown you have to create first a password file with ORPAWD utility, enable parameter REMOTE_PASSWORD_FILE=TRUE in initSID.ora, after doing this bounce the database grant sysdba or or sysoper to a user who you want to do the remote startup/shutdowns. Let's assume you grant one of those two privilege to user system.
This is server side.

In client side you do

connect system/manager@connect?string AS SYSDBA

once connected you can do remote startup shutdowns. You can only do this from SQL PLUS if client is from Oracle 8i.