Word in background with and text overlay

Another approach you could take would be to make an Adobe Express document with the text that you want to split (in this case, "POST"). Download it as a PNG. In your main document, go to Media, select Upload from device, and navigate to where you downloaded the PNG. Copy and Paste the PNG so that you have 2 copies. Drag the edge of the frame that containst the PNG to effectively crop the portion of the text that you don't want visible. Do the same for the other copy to hide what appears to be the middle of the word. 

Go back to Media and do a search for "drop shadow." There are a few different shadows that would work. Apply one to where the text is cropped. Apply the Mulitply Blend Mode. Duplicate it and use Flip Vertical to flip it.

Add your background and move it below the other layers. You may need to also apply the Multiply Blend Mode to your text as PNGs.

I can see the feature request in the user voice is marked "under review", which means it will probably be considered soon!
https://adobeexpress.uservoice.com/forums/951181-adobe-express/suggestions/46742428-adding-a-drop-shadow-to-objects
you can vote for it if you want it implemented!

As far as I know the only way to do that would be to duplicate your background image and crop it to a rectangle that simulates the transparent area, even then you will not be able to get the shadow underneath the rectangle* (I think they are working on shadows other than text shadows). So for the moment being you are stuff with creating this kind of design in Photoshop (or at least the bits that are lacking in Express) and upload them as .psd into the Adobe Express project (this is made easy by using librairies)

*you could technically create a duplicate of that rectangle, set it to multiply and apply a blur in the ajustments, but the blur itself applies to the image, and not the crop itself