Welcome to an addictive hobby!  

The anode-driver will need some rework because the NPNs (T1 thru T6) will 
only drive-out 4.3V (5v I/O from the ATMega, minus 0.7Vbe = 4.3V) as the 
circuit is shown. With multiplexing, you will need what's known as a 
'high-side driver', and often that is done with PNP or PMOS devices.

On the schematic, you would swap collector & emitter pins and change the 
NPN to a PNP.
Next, you would need to add a base-resistor between the digital logic, and 
the base. This is to limit the base-current to a proper value. For a 170V 
supply, a maximum of 4mA per tube, and a rough-guess of transistor beta of 
50, you would need 80uA of base current. That would be about 2Meg ohms for 
the base resistor. The exact value isn't critical.
Lastly, you will need a high-voltage logic driver to control the PNP 
transistors. When the PNP is off, the voltage seen by the logic-driver will 
'see' about 170V, which means regular logic gates *might* get damaged. In 
reality, the ESD protection of the 74HC238 would sink the 80uA into the 5V 
supply and turn all anodes on.

You could use an HV5530 to drive the PNP transistors, as well as the tube 
cathodes. Which means you can get rid of T8 thru T17. A single HV5530 has 
32 outputs.

Now, you dont need to display 0000 thru 9999 for a clock, so some cathodes 
can be left undriven. For example, on a clock only digits 1 & 3 need to 
display all 10 numerals. Digit 2 displays 0 thru 5, and digit 4 displays 
0-2. So you could drive all required cathodes for a clock with only 29 pins 
on the HV5530, in other words, direct-drive. So you can get rid of the PNP 
devices as well and make the design much simpler.

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