What you want to do is to transport the program from one system to the other one. Mainly it depends on your authorizations in both systems and on the settings of landscape, but in case you are authorized and the systems are both in the same transport domain, you can do the following (I do that kind of stuff every day):
On source system
- start transaction
SE09
- create Transport of copies, specify proper target system
- include your report there; the entry will be
R3TR PROG <name of your program>
- release Transport of copies
On target system
- start transaction
STMS
- go to the list of systems (little icon with Truck)
- pick the target system there
- you get the list of the transports ready for import
- click refresh and your transport should show at the bottom of the list
- point a cursor on it and trigger the transport
- if everything goes right, your entire report is now in target system