Yes it can support an SSD. The motherboard has SATA II support (max 300 MB/s) which will limit the transfer performance of SSD's a little. The fastest ones are even pushing the limits of the SATA III interface (600 MB/s). However you would still benefit from much faster random access times. Also, at a guess your 160Gb hard drive is probably not able to perform reads at even 100 MB/s so clearly if you use with an SSD that is limited to 300MB/s by the interface, it'll still be a very noticeable improvement.
For optimum performance from your SSD a fresh re-install would be best.
And yes those Radeon X1550's are rather dated now. An upgrade will help with things that require 3D performance. So games and so on. It won't help with general computer performance. As a good middle of the road upgrade, the GTX 750 & GTX 750 Ti are reasonably priced, very power efficient and will offer a good deal more performance in modern 3D titles compared to what you have now. But of course, it depends on what you want to do with the system and your budget.