How can I make a last-minute birthday cake without using artificial flavor or color?
The options are very wide. You are not telling us what kind of cake you have in mind, so I'll be making a few assumptions here. But once you start thinking outside the box that equals "birthday cake" with "cake with different colours of icing", a whole world of options opens up.


Birthday cakes need not be (multi-)coloured.
A one-tone cake that plays with texture can be very elegant, from luscious glossy chocolate ganache to swirls of light and fluffy frosting. You don't even need great piping skills (although it's a neat thing to master some day), heaping on the frosting and texturing into big fluffy "clouds" either with a spoon or a knife can be very beautiful and even my 7yo can produce very impressive results.
Find edible decorations for that pop of colour.
A bright red strawberry or raspberry on a white frosting, a few sprigs of mint or chopped pistachios for a touch of green. Some lemon or orange zest, perhaps. Even some jam or a fruit compote will do, put on top or served as a sauce on the side. Or go for a nature walk and find edible flowers. 





Choose non-edible decorations.
Birthday candles aren't edible, so you can add more non-edible elements. Little flags on toothpicks or even a small garland held up by two or three large skewers come to mind. Perhaps spell out "HAPPY BIRTHDAY DAD" on bunting? Print a photo of your dad and mount it on a skewer or two? I have in the past even used toy figurines as cake topper. As long as everything that actually touches the cake is food safe, you needn't worry. You can use some parchment or plastic wrap to create a barrier under a topper, if necessary. The flags in the picture below are made from origami paper, masking tape is very handy, too, because it needs no extra glue.