2 First of all, then, I urge that supplications, prayers, intercessions, and thanksgivings be made for all people, 2 for kings and all who are in high positions, that we may lead a peaceful and quiet life, godly and dignified in every way. 3 This is good, and it is pleasing in the sight of God our Savior, 4 who desires all people to be saved and to come to the knowledge of the truth. 5 For there is one God, and there is one mediator between God and men, the man Christ Jesus, 6 who gave himself as a ransom for all, which is the testimony given at the proper time (1 Tim 2:1-6).
Vv 2,6 constitute an Arminian or universalist prooftext. On this general interpretation, the scope of prayer is commensurate with the scope of redemption and/or salvation. We should pray for everyone because Jesus redeemed everyone and God wants everyone to be saved.
Suppose we grant that linkage. If so, that's actually an argument for limited atonement.
Take the purpose clause in v2, which either expresses the purpose or desired result of prayer for rulers. Why does Paul direct Christians to pray for rulers? Not for the ruler's sake (although that might be a fringe benefit), but for the sake of Christians. Pray for your rulers because their policies will, for better or worse, impact the lives of Christians who live under their rule. Pray for wise, benevolent rulers.
But in that event, this is not a summons to pray for all rulers. Rather, it's only a summons to pray for rulers under whose jurisdiction you live, work, and worship. Contemporary rulers.
Given that rationale, it would be pointless to pray for dead rulers. Pointless for 1C Christians to pray for King Tut, Nebuchadnezzar, Alexander Great, &c. They do not, and never will, rule over you.
By the same token, it would be pointless to pray for future rulers or distant rulers whose policies can have no affect on you at your own time and place. Paul doesn't intend 1C Christians to pray for Napoleon, Montezuma, Disraeli, Catherine the Great, Cardinal Mazarin, Sun Yat-sen, Chairman Mao, or Teddy Roosevelt. For the administration of future rulers or distant rulers has absolutely no bearing on the lives of 1C Christians.
(To be sure, Christian readers need to mentally update this command, but the same restrictions apply.)
Not to mention the absurdity of a command (on the Arminian/universalist interpretation) to pray for rulers you never heard of. How could you? You have nothing to go on. You can't even get started.
So Paul can't be directing Christians to pray for all rulers, but only some rulers. For the scope of the prayer is qualified by the purpose clause. If, however, the scope of the prayer falls well short of universality, then, by parity of argument, so does the scope of redemption and/or salvation.
One can resist the conclusion by denying a parallel between the extent of prayer and the extent of atonement, but I'm just discussing the Arminian/universalist interpretation on its own grounds.
And even if one were to deny the parallel, the purpose clause is still damaging to the Arminian/universalist interpretation. Although it employs the same universal quantifier ("all") that's a running motif in the overall passage (vv1-6), the force of the quantifier in v2 is clearly delimited by the purpose clause. So Paul can and does use that quantifier in a restricted sense in the very context of the overall passage.
More generally, the Arminian/universalist interpretation carries the tacit implication that we should pray for people we never heard of, people we don't even know exist. Pray for generic persons, persons who, for all we know, may or may not exist–in the past, present, or future. A dragnet prayer for anonymous people, for nonentities, just to cover your bets.
Typically, in Scripture, prayer is more personal and specific. You pray with someone in mind. You don't pray for someone who might possibly exist. You don't pray for blanks.
Even in corporate prayer (e.g. Dan 9: Ezra 9; Nehemiah 9), it's prayer for members of the community to which the supplicant belongs. Like a small town where everyone knows everyone else. It envisions specific sins. It envisions a people with a common history. A known history.
Likewise, you can pray for a specific situation, like a natural disaster. You may not know the victims, but you know the conditions. In that respect, you still know what to pray for. That, however, is very different than a prayer that's completely in the dark.