C# (Visual C# Interactive Compiler), 78 bytes

x=>{for(int i=35,c;(x[i]=(char)((c=x[i--])<48?c:c==57?65:c==70?48:c+1))<49;);} 

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Anonymous function that takes a char[] as input and outputs by modifying an argument.

Input is scanned from right to left and replaced using the following rules.

• The - character is ignored and processing continues
• The F character is converted to 0 and processing continues
• The 9 character is converted to A and processing stops
• The characters A-E and 0-8 are incremented by 1 and processing stops

C# (Visual C# Interactive Compiler), 77 bytes

x=>{for(int i=35,c;(x[i]=(char)((c=x[i--])<48?c:c==57?65:c>69?48:c+1))<49;);} 

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-1 byte thanks to @ASCIIOnly!

Anonymous function that takes a char[] as input and outputs by modifying an argument.

Input is scanned from right to left and replaced using the following rules.

• The - character is ignored and processing continues
• The F character is converted to 0 and processing continues
• The 9 character is converted to A and processing stops
• The characters A-E and 0-8 are incremented by 1 and processing stops

C# (Visual C# Interactive Compiler), 87 bytes

x=>{int i=35,c;do{c=((c=x[i])<48?c:c==57?65:c==70?48:c+1);x[i--]=(char)c;}while(c<49);} 

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Anonymous function that takes a char[] as input and outputs by modifying an argument.

Input is scanned from right to left and replaced using the following rules.

• The - character is ignored and processing continues
• The F character is converted to 0 and processing continues
• The 9 character is converted to A and processing stops
• The characters A-E and 0-8 are incremented by 1 and processing stops

C# (Visual C# Interactive Compiler), 78 bytes

x=>{for(int i=35,c;(x[i]=(char)((c=x[i--])<48?c:c==57?65:c==70?48:c+1))<49;);} 

Try it online!

Anonymous function that takes a char[] as input and outputs by modifying an argument.

Input is scanned from right to left and replaced using the following rules.

• The - character is ignored and processing continues
• The F character is converted to 0 and processing continues
• The 9 character is converted to A and processing stops
• The characters A-E and 0-8 are incremented by 1 and processing stops

C# (Visual C# Interactive Compiler), 88 bytes

x=>{int i=35,c;do{c=((c=x[i])<48?c:c==57?97:c==102?48:c+1);x[i--]=(char)c;}while(c<49);} 

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Anonymous function that takes a char[] as input and outputs by modifying an argument.

Input is scanned from right to left and replaced using the following rules.

• The - character is ignored and processing continues
• The f character is converted to 0 and processing continues
• The 9 character is converted to a and processing stops
• The characters a-e and 0-8 are incremented by 1 and processing stops

C# (Visual C# Interactive Compiler), 87 bytes

x=>{int i=35,c;do{c=((c=x[i])<48?c:c==57?65:c==70?48:c+1);x[i--]=(char)c;}while(c<49);} 

Try it online!

Anonymous function that takes a char[] as input and outputs by modifying an argument.

Input is scanned from right to left and replaced using the following rules.

• The - character is ignored and processing continues
• The F character is converted to 0 and processing continues
• The 9 character is converted to A and processing stops
• The characters A-E and 0-8 are incremented by 1 and processing stops