The apostle Paul is the pattern for all believers in this age (1 Cor. 4:17, 11:1; Phil. 3:17, 4:9; 1 Thes. 4:1; 1 Tim. 1:16; 2 Tim. 2:7, etc.). Therefore, we should pay close attention to each aspect of his life, e.g. his prayers. What Paul prays for is what we should be praying for, and, arguably, what Christ is praying for us above (Rom. 8:34), where the prayers are for believers.
Paul does pray for the lost as well (Rom. 10:1, 1 Tim. 2:1-4). He prays for lost Jews to be saved in Rom. 10:1 and for saved Jews ("the Israel of God") in Gal. 6:16 (mercy and peace on them). Note that both of these passages were written during the Acts period, when Israel had special gospel privileges under Paul's ministry (Rom. 1:16). So it may not be incumbent on us to pray for the Jews today in distinction from other men (1 Tim. 2:1-4), since their special privileges have passed, but these passages do show that Paul prayed for the lost, and so should we.