Is there a way to upscale without re-encoding?
To get from 1080p to 4K you're going to need to re-encode, but container won't matter (much), so 16bit png is overkill just for the sake of 4K.  H.264 works fine.

But before you spend the time to upsample, you should double-check to make sure that you actually need to.  Most modern 4K televisions upsample 1080p on the fly.  Most mobile devices aren't high enough resolution to display 4K.  If you can leave your project in 1080p and let the display device do the work, then you're already done!  

But if you're sure that's what you need, remember that normal, traditional methods of upsampling such as bicubic or bilinear interpolation don't really increase the actual resolution of the image.  Sure, the output pixel dimensions will change from 1920x1080 to 3840x2160, but the new image size just comes from (to over-simplify) doubling each pixel's width and height.  In other words, the basic scaling algorithms don't add any information to the image, they just make the 1080p "bigger."

There are newer, fancier upsampling algorithms which use sophisticated methods to scale images.  Photoshop calles theirs "Preserve details 2.0." DaVinci Resolve calls it "Super Scale."  After effects calls it a "Detail Preserving Upscale" effect, but they're all very similar methods of upscaling an image while preserving sharpness.  Bear in mind that none of these methods actually add information to the image, either.  They're just a tad better than ordinary, re-scaled 1080p.  They're also fairly computationally expensive.

In a perfect world, the best thing to do is to shoot in a resolution that's at least as high as your deliverable, but preferably higher, and then downsample.  When that doesn't happen, you'll have to evaluate the importance of having a higher resolution and weigh that against the extra time and effort it takes to generate such an image.  

4K is still pretty difficult to work with under many circumstances, in terms of bandwidth requirements, latency, storage, etc.  There's a lot of overhead that goes into those extra pixels, and it's not exactly a 1:1 relationship.  Quadrupling the number of a pixels means you can expect MORE than four times the effort to complete a project, especially if you're working with long timelines, multiple streams, multicam, effects, etc.