What is the rise time of an output pin of atmega328 changing its state?
Last time I measured on a circuit (it was an i2c network master and with a 2560), it was about 0.3us, so that gives you a high value for that. Also, I believe it cannot be faster than one/half of a CPU cycle which takes: 1/(16E6)=62.5ns/2 = 31.25ns, so here's the lower boundary. 

And there you got a possible range.

Then the rise time will definitely depend on the configuration of the circuit. It might be less or more depending on pull ups, capacitance attached at the pin out exit. It might also depend on the configuration of the pin (PWM? Internal Pull-up?…)

Finally, I believe there should some nominal values given in the datasheet.

Edit: Useful knowledge from the datasheet, but there's no definitive answer for any pin:

P311: 29.4.3 External Clock Drive

| Symbol | Parameter | Min | Max   | Min | Max   | Min | Max   |
| ------ | --------- | --- | ----- | --- | ---   | --- | ----- |
| tCLCH  | Rise Time |     | 2.0µs |     | 1.6µs |     | 0.5µs |


P313: 29.6 SPI Timing Characteristics

| Description    | Mode      | Min | Typ   | Max    |
| -------------- | --------- | --- | ----- | ------ |
| Rise/Fall time | Master    |     | 3.6ns |        |
| Rise/Fall time | Slave     |     |       | 1600ns |


P315: 29.7 Two-wire Serial Interface Characteristics

| Symbol | Parameter                      | Condition | Min        | Max   |
| ------ | ------------------------------ | --------- | ---------- | ----- |
| tr     | Rise Time for both SDA and SCL |           | 20 + 0.1Cb | 300ns |


HTH