Edit: On closer inspection, is that battery puffy? If a LiPo battery is puffing out, it is degraded and shouldn't be used.
No. You can't use that charger with your battery. As others have mentioned, that battery is a ticking bomb. On top of that, the charger probably has a charging current of 1000mA. Your battery has a capacity of only 90mAh, so at the recommended rate of 0.5C, that's a maximum charge rate of 45mA, which is well over 20 times less than the charger provides.
Very puffed = dangerous.
That's a very common battery charger board. You could use it to power a 3.3v arduino (like a 3.3v pro mini, etc) (connect output of that board to Vcc and Gnd, so the board will run at 3.7v (not 3.3v - but this is fine). It's a fairly small battery; you will probably need to put the micro to sleep most of the time to be happy with the battery life (and the usual removal of power light and regulator)If that battery were beefier, you could run a 5v board with a boost converter, but with a battery that size, you wouldn't be happy with the battery life. Edit: On closer inspection, is that battery puffy? If a LiPo battery is puffing out, it is degraded and shouldn't be used.
... don't it have a build in voltage converter? ...
I live in Israel, I hope I will be able to get the battery (because its chemical or something).
No it doesn't. Here is the datasheet. The circuit of your board is shown on page 3.I don't know about your country. There are shipping limitations on lithium batteries in the US, because there have been spontaneous ignitions in depressurized cargo holds on airplanes. They are pretty ubiquitous though. If you use this charger, you should get a larger battery though. Or replace the 'RPROG2' resistor as shown, to accommodate your battery's best charge rate.And yeah, use a lower voltage circuit if you want to use this battery.