Voice of experience here. I'm in Detroit and have manufactured plastic bumper covers for years. I used to injection mold Mustang bumper covers.
That front bumper cover is not rubber, although it is "rubbery" and people mistakenly call them "rubber bumpers." Your bumper cover is reaction injection molded Polyurethane plastic - or PU for short (sometimes PUR for polyurethane resin - PU and PUR are two different names for the same plastic).
Polyurethane plastic is very adhesive so you won't have any trouble repainting it. (Go to a hardware store and you will find polyurethane glue - two part resin glue is basically the same stuff - that's how adhesive it is.) You don't need to remove the existing paint, just sand it smooth then paint over. It will probably take you 45 minutes with 320 grit sandpaper. Primer paint any places where you cut through the paint to the raw plastic. Make sure that you add a flex additive to the paint (and use a flexible primer too) because the plastic bumper cover is flexible and without the flex additive the new paint will crack. DO NOT put adhesion promoter on the part as someone else here suggested. It will destabilize the bond between the existing paint and the plastic and a month later your new paint job will be flaking off. Adhesion promoter is only used on the newer style (since around 1995 and newer) TPO plastic bumpers (and PP and modified-PP and TPE just to be technically correct). Adhesion promoter is used on the raw TPO (et. al.) plastic as a bonding agent to help the primer paint adhere to the raw plastic because the adhesiveness of the TPO plastic is low compared to PU.
If you need more info on painting plastic bumper covers, you will find a paint resource guide here: