Do we preach in church? No.
But isn’t that what Paul tells Timothy to do? “Preach the word! Be ready in season and out of season….” No. The word translated “preach” there is kerusso. It means public announcement, not private commentary to an in-group. (Check the lexicon; do the word study in Scripture; expand the word study to the secular literature – all the evidence points the same way, as I’ve argued elsewhere.) It’s not something you do with an in-group in a home; it’s something you do in the marketplace at the top of your lungs for anybody in earshot. That’s just what the word means throughout the literature (notwithstanding our English misappropriation of it).
2 Timothy 4:2 is not an exception to the general usage of kerusso. Absent a compelling contextual reason to read the Sunday meeting into the passage — and it isn’t there — Timothy would have heard the word in its ordinary sense. The only reason we don’t hear it that way is because we’re imposing our usage of “preach” on the passage. Public proclamation was a mainstay of Paul’s ministry, and it’s not exactly a surprise that he charges Timothy to carry on this aspect of his work. The inclusio with “do the work of an evangelist” in v. 5 clinches it, if we needed additional evidence of its public-facing meaning.
Should we thunder the Word from the pulpit? Absolutely. Arguably, that falls under the biblical headings of teaching and prophecy, but in any case there’s not much exegetical case for calling it “preaching.”
But let’s look more closely at the context here. In chapter 1, Paul addresses Timothy’s qualifications and his inner life/personal prerequisites for ministry. He continues that theme in 2:1-13, challenging Timothy to endure hardship for the sake of God’s chosen people. 2:14 forward specifically addresses the way Timothy should minister to those people within the church community, and then (3:1ff) begins to address the hazardous people Timothy will face in that endeavor. Beginning in 3:10, Paul returns his focus to Timothy, contrasting him to the people in 3:1-9 and challenging him to continue in what he’s been taught, knowing that the God-breathed Scriptures themselves will fully equip him.
4:1 begins Paul’s final charge to Timothy, and here he begins with a command that specifically means public announcement and concludes in v.5 with “do the work of an evangelist.” As with his instructions for Timothy’s conduct within the church in 2:14-3:17, Paul leads off with the command (2:14//4:1-2), follows with a warning that it’s likely to be ill-received (3:1-9//4:3-4), and returns to Timothy with “But you…” (3:10ff//4:5). He follows the same pattern of instruction as when he was talking about Timothy’s ministry within the church, but this time, he’s talking about how Timothy faces the world.