First of all, it depends what you are comparing it to, and secondly, it depends what kind of item you are talking about.

If you compare most of Japan to prices in Manhattan, it's probably about the same.  If you're comparing the prices to the prices in the middle of Iowa, then you're goign to be in for a shock.  Another thing to note is that the quality of Japanese good is typically higher (And Japanese income tax is lower than most countries, leaving more money to spend), so there are reasons why you will pay more.

Many Americans muse "Japan is so Expensive", but everyone becomes used to the prices in their home market and thinks of them as "normal".  People in Mexico and China also think the US is expensive.  As a general rule, expensive things are the same price or cheaper in Japan, while things that are dirt cheap are much more expensive in Japan because of the high wages and quality inspections, etc.

Another thing to consider is that there is less competition in Japan in general, both domestically, and with international brands.  Import taxes are high, and usually only a few large conglomerates produce things in Japan.

Alcohol:  Alcohol is almost without exception cheaper than most countries, especially the US.

Tobacco: The same story as alcohol.

Apartments:  This, as anywhere, depends on various factors, maybe location.  Because Tokyo is spread out, it's cheaper on average than London or Manhattan.

Real-Estate:  The prices for buying are not cheap, but the interest rates of less than 3% on mortgages make purchasing much more attractive if you can get a loan.

Food: Groceries are expensive, but high quality.  Some things aren't too bad, other things (like cheese) are.  Expect to pay about what you would pay at an expensive organic grocery store overseas.  Certain specific items (some kinds of grapes and melons) are crazily expensive.

Electronics Appliances: In the same neighborhood as in the USA.  Many people expect these to be cheaper since they are often manufactured in Japan, but the reality is that many are made in China these days anyway.  Also, the quality control is much higher, fashion (having the latest model) is much more of a concern to people, and a recycling fee is included in the price of many major electronic items.  Foreign made goods (i.e. anything Apple) will cost significantly more.  On a positive note, there are many things electronic which are available only in Japan, or at least available here first.

Electronic Parts: Things like memory cards can be found vastly cheaper in Japan than overseas if you know where to look.  That is to say, you can get mail-order prices in person.  

Cars:  Cars aren't cheap, and besides the purchase cost, gas isn't cheap, insurance isn't cheap, inspections aren't cheap, getting a license isn't cheap, and if you live in the city, parking is going to cost a pretty penny too.

Train:  It is said that the train in Japan is expensive, but I don't find this to be the case.  Anyone who lives within 30 minutes of work can usually commute for less than USD $100 a month if they have a commuter pass, which is not bad I think (and certainly cheaper than a car would be - and many companies won't allow you to come in your personal car for liability reasons(.

Furniture, etc.:  Again, the quality focus makes things expensive, but firms like Ikea are here now, and "recycle shop" second hand stores have like-new items at good prices.

Internet Access: Very cheap when you consider the speed.  Currently one can get 100 or 160 Mbit internet service in major metropolitan areas for around USD $40 per month.  You can also swap that for 21 Mbit use-anywhere 3G service instead for around the same price, if you have a laptop.

Medical Costs: Medical care in Japan isn't cheap, but there is government funded medical care, which means if you sign up, you only have to pay 30% of any expenses incurred.  (Within certain limitations, of course).