Almost, but you'll need to include some diodes to stop cross-linking of segment combinations... The schematics below show what happens in a simplified case, where we have a 2-position switch and 3 LEDs - a, b and c. With the switch in the left hand position we want LEDs a and b to be lit, and with the switch in the right hand position we want LEDs b and c to be lit.
Note here that, because LED b is common to both switch positions, it forms a link between both sides of the switch, and so even though the switch is in the left-hand position we will get unwanted current flow through the common link (as depicted by the red arrow) and across to LED c which will then also be lit.
If we now include diodes in series with the connections between both sides of the switch and LED b, this cross-link is broken - these series diodes prevent current flow from one side of the switch to the other.
For this example, the series diodes on LEDs a and c aren't necessary, because neither of these LEDs are shared between switch positions, however for the full circuit you're intending to build, you'll find that all of the segments are shared between at least two of your switch positions, and so you will need these series diodes on each LED.
HTH
Chris