Brewster's Millions (1985)
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First, there's no indication of just how much time has passed since Rupert found out about Monty and when he filmed his will. Rupert could have had private investigators or the like searching for his heir for years or months before he filmed the will &, with the fortune he'd amassed, he could have gotten very fast results by hiring a lot of people to do so. When he was informed by one about Monty & what kind of life he'd led, Rupert then ordered the filming of his will.
Secondly, the end of the month where Rupert died could have been merely a few days before Monty saw the film. The way Roundfeld described the circumstances of Rupert's death were just a little under-detailed but we get the idea. Rupert had known about Monty for some time -- putting together an estate and a forming a will takes a bit of time, even for a rich guy like Rupert. He likely hired Granville & Baxter because they were near where Monty was living and they were high-profile in the New York City area. Or Rupert could have worked through an office they had where he lived himself or an affiliate office of theirs. Edit
It does seem ridiculous that Rupert could talk directly to Monty after he died and even answer Monty's questions like he was in the same room. It's simply an old trick used by the writers to make the scene funnier, akin to an old comedy technique called "breaking the fourth wall" where a character on television or in a movie speaks directly to or glances at the audience. Hume Cronyn's lines were written in such a way as to make him seem more comical.
More on the fourth wall can be read here. Edit
Monty had only rented the clothes, the apartment and the lush office that he does business from (the one that Margaret spends all her time decorating). Monty may seem like a regular guy who doesn't know how to manage money but Granville & Baxter severely underestimate Monty himself, who turns out to be much more savvy. By renting everything from high-end companies at high prices, he can wipe out much of the $30 mill.
The car that he buys for Angela is an Aston-Martin V8 Volante, about 1985 and could be considered an asset: in the film Rupert specifically says Monty can't "buy a Hope Diamond for some bimbo for her birthday" which means Monty can't just buy the car & give it to Angela. The scene was probably written the way it was to give the audience a scare, thinking that Monty, being so infatuated with Angela, might be willing to break the terms of the will just to be with her. However, he could have just been renting the car for her too, tho it seems highly unlikely. Edit
It's not. Monty merely used it to send a postcard to Baxter & Granville as a way of rubbing their noses in it. The stamp is not destroyed -- it was canceled by the post office & severely devalued. If Monty had taken it home & burned it or otherwise destroyed it, then it would be a violation of Rupert's conditions. By sending it to Baxter & Granville and devaluing it, it becomes their property and virtually worthless.
Collectible & extremely valuable stamps are prized by collectors the most IF they haven't been circulated, IE, they hadn't been used for postage. The stamp in question is indeed a valuable one, the "Inverted Jenny" and is highly prized by collector's or other people who like to own valuable merchandise. If you look in a stamp collecting book that lists the value of rare stamps, there are several columns with monetary values for each stamp. One column, usually the one on the far right, will be the value if the stamp is in "mint" condition: hasn't been used for postage & shows almost no signs of wear. An argument could be made that the canceled stamp given to G&B could still carry some value but the writers have obviously glossed over the details to make the scene both funny and dramatic and to show that Monty is a lot craftier than G&B assumed.
One must also keep in mind that it is a postage stamp, so its originally intended use is to be used as postage to mail something. Thus he was using the stamp for its originally intended use and this is NOT the same as deliberate destruction of said property.
More info on the Inverted Jenny can be read here. Edit