Pass command line parameters to a program inside the shell script

The usual way would be to save a copy of arg1 ("$1") and shift the parameters by one, so you can refer to the whole list as "$@":

#!/bin/sh
arg1="$1"
shift 1
/path/to/a/program "$@"

bash has some array support of course, but it is not needed for the question as posed.

If even arg1 is optional, you would check for it like this:

if [ $# != 0 ]
then
    arg1="$1"
    shift 1
fi

You can slice the positional parameters using parameter expansion. The syntax is:

${parameter:offset:length}

If length is omitted it is taken as till the last value.

As you were to pass from second to last arguments, you need:

${@:2}

Example:

$ foo() { echo "${@:2}" ;}

$ foo bar spam egg
spam egg