Yes, you can use your measured resistor.
Your schematic does not specify a resistor stamped with a 1% specification or with a gold band, but a resistor with an actual value within 1% of the value indicated on the schematic.
If you purchased a 1% resistor, you can rely on its indicated value and should not have to verify the actual resistance value yourself. If you purchased a 10% resistor, or wanted to reuse an old resistor with all color bands worn off, you should measure it first.
Regarding stability: simply writing 1% on the schematic does not imply that your device will have to operate for X nr of years within spec. Nor that it will operate over a wide range of ambient temperatures . . .
That said, I would like to add that in lots of cases the values of resistors are not critical. If someone calculated that a resistor of 1000 ohm would work fine, (s)he might specify a 1% because this does not cost more than a 10% one, if you could get one. But there is no promise that the device would not work with a 1011 ohm resistor. One of 1500 ohm might work just as well, perhaps even better, that is completely dependent on the design. A designer might specify all resistors of the same value of 1000 ohm, even if other values might work just a little better, because it is easier to have lots of equal parts. Again, it depends on the design. Use your own skills and judgement and don't let you be intimidated by the "1%" label.