I can't precisely answer your question. It depends on what the new assessment is, what the old assessment was, what your tax rate is, and how much you paid for the townhouse.
If--and it's a big IF--your assessment is supposed to be 100% of market value, here's how to figure it out.
Check what the previous owner's assessment was and what his taxes were. You can probably find both on the listing sheet for the property. If not, you probably can find it online at your tax assessor's office.
Let's make up some numbers here. Let's say the property was assessed at $100,000 and the owner paid $1,000 in taxes.
Now, let's say you bought the property for $110,000. Most jurisdictions will adjust the tax assessment to a recent sale. After all, if the assessment is supposed to reflect the value of the property, what better way to determine its value than what someone just bought it for?
From there, it's simple math. $100,000 is to $1,000 as $110,000 is to "x." In this case, the assessment would adjust up by 10%. Therefore, the taxes also would go up by 10% . . . to $1,100.
Or you can look at the tax rate. In our example, it's $1.00 per $100 of assessed value. So if the assessment goes up by $10,000, then the tax itself would go up by $100.
Two different ways to get to the same answer.
Most jurisdictions will adjust the assessment periodically--anywhere from once a year to once every 3 years or so. So, just because the last owner lived there for a long time doesn't mean that his taxes stayed flat. They probably adjusted up and down based on general market conditions.
Another variable is that the taxes you pay are based not only on the assessment, but also the tax rate. And that can change yearly, too. In the example above, the rate was $1.00 per $100 of assessed value. But next year the city council could decide to adjust the rate to $1.05, for instance. In that case, even without a change in assessed value, your taxes would go up $50.
The main thing to look at, though, is the difference between the property's current (old) assessment and what you paid for it.
Your Realtor should be able to help you with all that.
Hope that helps.