Without knowing you or your unique circumstances it is impossible to give a fully appropriate answer. However, in response to the primary question being asked: It would not be permissible to use an iPhone on Shabbat for these purposes.
Firstly, while technology can be a tremendous blessing, it is also causing people to turn to devices for tasks that we used to be able to do on our own. Instead of relying on your iPhone to recite prayers and study Torah, or even a Siddur and Chumash for that matter, since you have no other choice on this particular Shabbat, why not do your best to say the prayers that you know by heart and discuss what you remember about the Torah portion? After all, though it is ideal to recite all of the prayers, there is still value in reciting just some, such as the Shema.
Secondly, although it sounds noble to use an iPhone for this purpose on Shabbat, you would be putting yourself into a tempting situation to check Email, go online or make phone calls with just one touch of the finger. Since this device is used all week long for such activities, one may easily slip into them on Shabbat as well, without much thought, thereby going from using the device for noble and holy purposes to using it for mundane weekday reasons. Much of the beauty of Shabbat lies in the fact that it gives us a respite from the swiftness of our daily pace and other obstacles to meaningful interpersonal and spiritual connections. Using such a device on Shabbat may appear to have valid reasons, but is not in consonance with the spirit of the day, and can easily slip into activities which are certainly not suited for the holy Sabbath.
Thirdly, doing what we consider to be a Mitzvah via forbidden means is similar to the Talmudic concept of a “Mitzvah Haba’ah B’Aveirah” (a Mitzvah that is brought about through a transgression). Therefore, while it is laudatory to want to pray and study Torah, since doing so in this manner would be forbidden it would not fulfill a Mitzvah.
While it is very easy to rationalize numerous technologies that might seem to enhance Shabbat, the best way to truly carve the Sabbath into a sanctified island of time is not to rely only on our own judgment about what actions seem appropriate. Personal needs or emotions often cause us to mislead ourselves into rationalizing behaviors that are not actually conducive to experiencing the sanctity of Shabbat. However, by following communal norms and traditional rabbinic guidelines, we are enabled to experience true transcendence and go beyond ourselves, ultimately connecting to the Divine. Therefore, you should not use your iPhone for this reason on Shabbat, and I believe that in the long run you will be much better off for it.