upgrade alarm to remote start?

Yes, absolutely.  You can even buy remote starts that will run right off a car's factory keyless entry.  But the results can vary.

There are a few extra issues the installer will have to deal with, namely, re-wiring a few things to keep the alarm from going off during remote starting.  They could wire in a relay to interrupt your alarm's shock sensor, but more often than not, they just unplug the sensor or turn it all the way down.

Also, if it's set up to trigger off your existing alarm's remote, the range will suck.  Or, you'll have to carry two remotes around.  If you buy the alarm and remote start from the same brand, a good installer can program one remote to operate both units directly, which would be best.

So, you can do it, but it's a pain, and it will probably cost you a little more in the end...better to just buy a combo unit all at once.