Okay, imagine you have a smaller display, 3 rows left and right, and 4 columns up and down.
To make dot C0RO flip black to yellow, you need to drive Column0 High and Row0 How.
To make dot C0R0 flip yellow to black, you need to drive Column0 Low and Row0 High.
Same for other dots, say C2R2 - drive Column2 High, Row 2 Low, or Column2 Low and Row2 High.
ULN2803 can only drive its outputs low, so something more is needed.
To be able to drive current in both directions, you need an H-bridge IC, such as this
http://www.digikey.com/product-detail/en/TA7291SG(O,J)/TA7291SGOJ-ND/1730027
http://www.toshiba.com/taec/components2/Datasheet_Sync/200706/DST_TA7291P-TDE_EN_3604.pdf
Tho it is not exactly clear from the picture on sheet 2 of the datasheet (which has 3 sets of pins shown for 3 different package types), here’s how it works:
When IN1 is High, current will flow from Vs thru the top-left transistor, thru the motor M from top to bottom, thru the bottom right transistor, to Gnd.
When IN2 is High, current will flow from Vs thru the top-right transistor, thru the motor M from bottom to top, thru the bottom left transistor, to Gnd.
See the Function Table on Page 3 as well.
So you need to work out if your card is wired up similarly, and if so, you can work out the connections for 42 H-bridge chips. Or 1 chip and a bunch of relays. Or something else.