Why was Christianity such an appealing alternative to other religions during the period 500 BC - AD 500 (Ancient Rome and Early Christianity)?
When answering this question, the first thing to do is to understand that the dates on this question do not match with the question. Christianity could not have been an alternative to other religions at any time before the Common Era. This is because Christianity did not exist at all until after the death of Jesus around 33 CE and it was not a large religious movement for some time after that.
However, once Christianity did come to exist, many people in Rome saw it as an appealing alternative to other religions. We cannot know for certain why they did so. Christians might argue, for example, that it was an appealing alternative because it was true while other religions were not. Historians can only make their best guesses as to why Christianity spread.
One reason often given for the success of Christianity was that it was a religion that emphasized a caring god. The Roman religion did not give its believers any sense that their gods actually cared about them. Their relationship with their gods was more transactional; if they did certain things, their gods would do certain things in return. By contrast, the Christian god was said to care about his people for themselves. This might have made people feel that Christianity was more emotionally satisfying than other religions.
In addition, Christianity provided a clear moral code for its adherents to follow. The Roman religion did not do this. The gods did not demand that people should act in any particular way towards one another. Christianity, of course, was very concerned with the way in which people behaved. This might have made Christianity more appealing because it would have helped people believe that they knew the right way to behave.
Finally, Christianity was an inclusive religion with a message of brotherhood and equality. The Roman world was rather stratified and members of the lower classes in particular could feel abused by their society. Christianity taught that all people were equal before God. This might have made people from the lower classes (and women in particular) feel that Christianity was a better religion for them.
All of these are reasons advanced by scholars to explain why Christianity appealed to many people in the Roman world after the death of Jesus.