The protagonist has unlimited gold but needs to parlay it into financial capital, social capital, and political capital in order to effect change at the national level. It might be useful to think of tactics as being split into the carrot and the stick.
The carrot (a.k.a. https://en.wikipedia.org/wiki/Soft_power) involves encouraging others to have the same goals as the protagonist and is more likely to win allies. This can include:
- Indirect (e.g. doing things that make their domains more wealthy or powerful without their knowledge) support to nobles and local leaders who are sympathetic to the protagonist's cause. Unless there's a crisis or other urgent need by the recipient, direct support (e.g. gifts) will correctly be seen as attempting to buy influence and may backfire.
- Find other nobles who are neutral to the protagonist and his cause but are enemies of his opponents and provide indirect support to make the neutral powerful enough to take down the opponent. (But bear in mind that nobles support each other; if the neutral later turns against the protagonist, they may find that they solved one problem by creating a bigger one down the road.)
- Direct charitable donations or indirect support to religious leaders or religious orders whose ideals are close the protagonist's. Remember that religion is a profound influence in medieval times. Ideally, the religious organizations would openly declare slavery / corruption / etc. to be immoral. Again, beware of the appearance of giving out bribes.
- Create deals or ties to the great merchant or banking dynasties and use that influence to carry out their goals. Having infinite gold means that business risk is eliminated, which is an unparalleled advantage.
- Supporting organizations or individuals within them (e.g. guilds, chivalric orders, learned societies) who support the protagonist's ideals. If the protagonist can find the right person, they can even create new such organizations who have policies that are pro worker/anti-corruption as part of their charter.
- Leading by example. The protagonist can navigate their way into de jure or de facto control over a region and demonstrate to the nobility that their policies lead to increased wealth / productivity from the population. The nobility might not care about morality or ethics but they do care very much about income.
- Win influence by doing good works, e.g. creating orphanages, helping the poor, etc. A very common trope in isekai novels.
- Providing technological boosts. Already discussed in other comments.
- Become the supplier or manufacturer of something that can be provided to the domains of friendly nobles that gives them a military or economic advantage over their rivals.
The stick (a.k.a. https://en.wikipedia.org/wiki/Hard_power) involves coercion. Direct confrontation is likely to be disastrous as discussed elsewhere but covert options are plentiful such as buying up scarce resources their opponents need, financing sabotage against the domains of hostile nobles or hiring away talented people from them, financing espionage to expose corruption, financing covert attacks on slave trading organizations (no slave traders left alive = no more new slaves), financing propaganda or rumor campaigns to stir up public sentiment against corruption, organizing and financing slave escapes / revolts, etc. Covert means covert; discovery of coercive tactics by the protagonist's allies may turn them against the protagonist.
Whichever tactics the protagonist uses, they should not expect this to be cutely bloodless, as it is in isekai novels. Even if they're profiting on corruption, slavery, oppression, etc., a noble house is an extended family and will counterattack swiftly and viciously as a whole if their livelihood or prestige is threatened. All of the tactics above can be equally be used against the protagonists and their allies and they have the advantage of being long-established natives.
Some prerequisites also should be borne in mind:
- The protagonist must give considerable thought into inserting themselves as a respectable member of society in a way that disguises/protects their source of gold, unless they want to spend the rest of their lives in a dungeon being a goose laying golden eggs for someone else's benefit. They might, for example, be denounced as an agent of a foreign power or even an agent of an evil deity.
They cannot count on anonymity; medieval communities are small, the merchant/banking community is smaller, the nobility is even smaller still, and anyone spreading money around in large quantities is going to be remembered. One way they might launder their magically gotten gold is to pretend to be a merchant from a far distant land who negotiates or buys their way into an impoverished merchant house in their target kingdom.
- Second, the protagonist needs intel, both through personal observation and a network of people providing reliable information on the state of the world. A traveling merchant may be a way to visit various cities to find out what the culture is like, who's in power, what factions exist, what their views are, what their strengths and weaknesses are, etc. as well as the views of the common folk on their lot in life and the reforms. (Your protagonist may be unpleasantly surprised to find that the commoners don't actually care much about reform on moral grounds unless they themselves have been victims.) As part of the travels, the protagonist will probably want to meet and make friends with other merchants and townsfolk of influence, both to build their intelligence network and future power base.
- The protagonist and their allies need protection against direct attack. In isekai novels, this is usually fulfilled by the attacker being ridiculously powerful themselves or being protected by their, ahem, bosom buddies so to speak. Lacking either of these, hiring security / troops is an option for your protagonist but it's worth remembering that they work for a paycheck and therefore have no personal loyalty to the protagonist and won't fight to the death for them or can be bribed away.
- In order to have allies to support, the protagonist needs to gain allies in the first place. This will require them to have absurd levels of political acumen and interpersonal skills to both proselytize their cause and determine who will support them and who just wants their gold.
tl;dr The protagonist is taking on a rather crummy task that requires them to be ridiculously overpowered, albeit in different ways than usual, and will probably take a lifetime or more to achieve.
