How would he even regain any honor by winning? Those things don't have anything to do with eachother.
I don't think Alec is wanting to gain any personal honor but is instead looking to gain some of his family's honor back after the mess that Jilvan (Alec's younger brother) and Geier caused by trying and failing to cheat during the Knight selection test. More specifically, Alec thinks that by beating Yukki (a lowly Baron's son who is below Alec's family's standing as the Marquis's son) he will gain some of the respect from the Nobels that his family lost due to the fact that his brother lost the knights selection test twice (once by cheating (having Geier summon a ghost) and secondly by being chocked out by the ghost who possessed Geier (after he summoned it)). What I find truly hilarious is that Alec really thinks that he could win against Yukki having never seen him fight in a battle. Currently, there are several rumors going around about Yukki's accomplishments in battle (saving the princess from bears, killing a griffin, and defeating the ghost in the Elysion) and to not take into account any of them I feel is just foolish.
I believe that Alec is thinking that if he wins the duel then he will have put the lowly Barons Bastard son in his place and he will have accomplished two things: first, he will regain the respect of the nobles that were lost by his younger brother's actions during the knight's selection test and secondly, he will have shown that Salvia (Iris's sister from the main line of ascension) has much stronger support (the Marquis family) than Iris and she is a smarter princess because she chose a knight who was stronger than Iris selected.
One last thing. I wonder if Alec has considered what would happen if he lost the duel. Often in these types of stories, you will see a side story of how Alec ended up as Salvia's Knight (He is taking responsibility for his brother's dishonor, his family forced him to become Savia's knight, or Iris's family (either the King, Queen, a brother or a sister (more specifically Salvia)) felt she was gaining too much support (being a bastard child herself) with her selected knight and they stepped in and brought the Marquis older son into the equation). Just a lingering question that I have.
Also, I would like to know if Iris was always being bullied by Salvia or if something happened and Salvia suddenly changed and started bullying Iris. I ask this because those around you can influence how you see and think of others. Just like when Kachel was manipulating Zeros to hate Yukki.
Sorry so long-winded