Andy saved Hadley roughly $24,000 to 25,000 in taxes.
Hadley's relative left him $35,000 from an estate totaling around $1 million. In the 1950s, estates valued $60,000 and under were exempt from estate taxes, so this estate is not exempt and Hadley would have to pay taxes. The federal estate tax rate in the 1950s for all bequests of $10,000 or more was 77%. Therefore, Hadley would have had to have paid $26,950 of his $35,000 bequest to the federal government, leaving him with $8,050. It is possible there would have been additional taxes if Maine (where Hadley lived) or Texas (where the deceased relative lived) had estate taxes at the time. Maine currently does, but I can't find information about Maine in the 1950s.
In this scene, Andy states that the federal government has a gift tax exemption up to $60,000; this is incorrect. In the 1950s, the federal government exempted gifts from taxation only up to $30,000. Therefore, Hadley could gift $30,000 to his wife and the federal government could not touch it. States do not tax gifts.
This leaves Hadley with $5,000 of his $35,000 bequest remaining that he will have to pay a federal (and possibly state) estate tax on. I cannot find tax rates for bequests lower than $10,000, but we can assume it would be something lower than 77%. Depending on the exact rate, it is likely that Hadley would have to pay as much as $2,000-3,000 in estate taxes on the unexempted $5,000. Assuming a $3000 (60% rate) hit on the unexempted portion, Andy saved Hadley $23,950 in taxes alone, possibly more. Hadley would keep probably in excess of $32,000 instead of just $8,050. As a point of reference for what this might mean to Hadley, California prison guards in 1957 made a maximum of $436/month ($5,232/year).
Sources for historical tax rates and exemptions are below.
Tax Foundation source.
Lawyer fees would have been minimal compared to the estate tax savings, but Andy has reason to suggest they would be significant
Lawyers charge an hourly rate for filing tax paperwork, and it's likely Hadley would be billed for at least one hour to complete and file this paperwork on his behalf. The lawyer might charge him two or even three hours, but it's hard to imagine they would charge more than that. I cannot find historical rates for Maine lawyers, but in 2022 you can expect to pay between $89 to $326 per hour for a lawyer's time in Maine. So at today's rates, Hadley would at most pay $1,000 in legal fees (probably closer to $300), and that's before you adjust for the fact that the story occurred 70 years ago. Hadley's fees would be far lower, likely less than $50.
However, Andy suggests that the lawyer fees will be expensive: "And that'll cost you." This is part of Andy's ruse. He want's Hadley to think he'd get ripped off by the lawyers so that he'll give Andy and the guys beer for Andy to do the paperwork. That's why he misleads Hadley into thinking a lawyer would be so expensive.
In the grand scheme of things, a few six packs of beer would be nothing
The beer would be a drop in the bucket of course (no pun intended). A 6-pack in the 1950s was less than $2.