How much difference is there between seat height and handlebar height? Are the bars lower than the seat when the bike is level, or are the bars higher than the seat. There are two ways to get more upright - move the bars higher, or move the bars closer to the seat. Both approaches have their advantages and disadvantages. Getting the bars higher may be much less expensive than a conversion to flat bars.
Maybe all you need is an "up-jutter" stem and maybe new cables to get the bars a little higher.
A flat bar conversion will likely require new bars, new stem, shifters and cables. It will also have some effect on the handling of the bike.
I'd try raising the bars first.