Unfortunately (or fortunately, depends on how you see things) you can get in the ballpark but getting colors that saturated is hit and miss with digital. Depends on the light you're shooting in. It's easier around sundown than midday. Also, using a polarizer filter helps but often won't quite get you there, either.
One reason is that Kodachrome did during processing with chemicals what Photoshop does now during post-processing with curves, adjustment layers and blending modes. If you look at the photo and look around you during the same time of day the photo was taken (late afternoon or late morning), the colors you see in real life are seldom that saturated. Sometimes close, but seldom to that level. That extra saturation was added by the film manufacturing and processing chemistry.
In addition, digital sensors have linear response to light, not the curved response that film had and that the human eye tends to have. That's why digital raw negatives often look flat compared to the actual scene.
One way to pop colors more in JPEG is to bump up the color temperature of your camera in your shooting menu. There should be a setting that allows you to make the color warmer or cooler. Making it warmer will help with this but again, due to the above factors you may not get to where Kodachrome was, although you could create an procession action in Photoshop or a preset in Lightroom to get the color saturation you want in either product.