To such contracts, a company's standard Terms of Service (ToS) apply, and by contracting with them, you agreed that those are part of the contract. Your initial contract did contain a clause that you pay for shipping, and your original total came out to X USD.
The Shipping turned out to be much more costly, possibly because Japan Post currently does only very limited shipping globally and other global carriers like UPS are more expensive.
Now, you have to read the ToS carefully, as it might contain a clause to adjust shipping prices. Most likely their ToS contains a clause to the effect that shipping prices are your responsibility in any case and that errors in shipping fees are to be paid by you. Then, you said they sent you an information mail that contained what could be seen as a re-negotiation of the contract, but it might also just be a notion about the adjusted shipping fees as required in the initial contract. Such clauses are generally legal.
In either way, shipping and transport costs are usually not costs that the company can choose on its own, they are dictated on them by the transportation company, which can pretty much dictate prices in any way they want. Especially international shipping can be extremely volatile: Sending a normal letter to the US from Germany incurs lower fees than some large national letters. However, shipping fees from the US to Europe start at about 18 USD for uninsured parcels and very quickly go up to more than 100 USD for the same shipped item but a different class of parcel. If the carrier decides your shipment is in a different class of parcel than anticipated, then the shipping company can do nothing but ship in that category.
Did they adjust the bill on the box?
While the actual shipping fees are now higher, it is more likely they did not adjust the bill on the box, if the fee adjustment was due to the carrier's insistence in the outgoing station. In that case: Lucky you, because the customs office uses the declaration on the box as the basis for customs and import fees - for which you are solely responsible. These fees are based on the value of the contents plus the shipping fee! Yes, you pay customs and taxes on shipping internationally. In case they did adjust the declaration bill as is proper, you'll also get charged more for importing, depending on the customs and fee schedule.