JavaScript (Node.js), 61 55 54 bytes

Saved 1 byte thanks to @nwellnhof

Takes input as an array of characters.

s=>s.map(p=c=>r+=p>(p=(+c?~c:1-Buffer(c)[0])%6),r=0)|r

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How?

For all characters but digits greater than $0$, the 0-indexed column $x$ is given by:

$$x=(c-1)bmod 6$$

where $c$ is the ASCII code of the character.

For positive digits $n$, we need to do instead:

$$x=(n+1)bmod 6$$

Examples:

"a" --> (97 - 1) mod 6 = 96 mod 6 = 0
"b" --> (98 - 1) mod 6 = 97 mod 6 = 1
"0" --> (48 - 1) mod 6 = 47 mod 6 = 5
"3" --> ( 3 + 1) mod 6 = 4 mod 6 = 4

Commented

s => // s = input string (as array)
 s.map(p = // initialize p to a non-numeric value
 c => // for each character c in s:
 r += // update the result r:
 p > ( // compare p with
 p = ( // the new value of p defined as:
 +c ? // if c is a positive digit:
 ~c // -(int(c) + 1)
 : // else:
 1-Buffer(c)[0] // -(ord(c) - 1)
 ) % 6 // apply modulo 6
 ), // yields 1 if the previous value is greater than the new one
 r = 0 // start with r = 0
 ) | r // end of map(); return r

Jelly, 11 bytes

⁾04yO‘%6<ƝS

A monadic Link accepting a list of (uppercase) characters.

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How?

First replaces any '0's with '4's (so the rest of the code treats them as being in the rightmost column). Then casts to ordinals, adds one and modulo's by 6 to get 0-based column indices. Then compares neighbours with is-less-than and sums the result.

⁾04yO‘%6<ƝS - Link: list of characters e.g. "BLAST20"
⁾04 - list of characters = ['0', '4']
 y - translate "BLAST24"
 O - ordinals [66,76,65,83,84,50,52]
 ‘ - increment [67,77,66,84,85,51,53]
 6 - literal six
 % - modulo [ 1, 5, 0, 0, 1, 3, 5]
 Ɲ - neighbourly:
 < - less than? [ 1, 0, 0, 1, 1, 1 ]
 S - sum 4

Perl 6, 45 39 bytes

sum .[1..*]Z<$_o(2 X-.ords)X%46 X%6

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Works with uppercase letters. (2-ord(c))%46%6 computes the reversed x coordinate.

K (ngn/k), 32 31 bytes

+/>':(+6 6#,/"a10"+!'26 9 1)?0+

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Clean, 85 bytes

import StdEnv
(s=sum[1\a<-s&b<-tl s|b>a])o map(e=(toInt e+if(e-'1'>'/')1 -1)rem 6)

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Ruby, 56 bytes

->sw=9;s.count

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Preliminary naive version, will be golfed.

Japt -x, 14 bytes

®rT4 c Ä u6Ãä<

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Port of this Jelly answer. Takes input as an array of characters, with letters uppercase.

Explanation:

®rT4 c Ä u6Ãä< :
® Ã :Map each character through:
 rT4 : Replace 0 with 4
 c : Get the char-code
 Ä : Increment it
 u6 : Modulo 6
 ä< :Replace with 1 if you had to move right, 0 otherwise
 :Implicitly sum and output

Java (OpenJDK 8), 73 bytes

Not a bad solution for Java! That zero being on the right-hand side cost me several bytes.

t->int a=9,c=0;for(int d:t)c+=a<(a=(--d+(d/48==1?2:0))%6)?1:0;return c;

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Explained

t -> // Lambda taking a char array as input
 int a=9, // Initialise last column value
 c=0; // Initialise frustration count
 for(int d:t) // Loop through all chars in title
 c+= // increment the frustration count if...
 a< // The last column is smaller than the current column
 (a= // Set last column to current column
 (--d+ // Decrement ascii value of char
 (d/48==1 // If ascii decremented ascii value is between 48 and 95
 ?2:0) // increment by 2 (1 total) or 0 (-1 total)
 )%6) // Mod 6 to retrieve column index
 ?1:0; // Increment if to right hand side
 return c; // return calculated frustration count

05AB1E, 12 11 bytes

-1 byte thanks to @Kevin Cruijssen

¾4:Ç>6%¥1@O

Another port of Jonathan Allan's Jelly answer. Takes input in uppercase.

Explanation:

¾4:Ç>6%¥1@O //full program
¾4: //replace all '0's with '4's
 Ç //get ASCII code points
 > //increment
 6% //modulo 6
 ¥ //get deltas
 1@ //is >= 1
 O //sum

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Retina 0.8.2, 46 bytes

T`l1-90`1-61-61-61-61-61-6
.
;$&$*
&`;(1+);11

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T`l1-90`1-61-61-61-61-61-6

List the alphabet and digits in the order on the OSK and map each one to a (1-indexed) column number.

.
;$&$*

Convert each column number to unary.

&`;(1+);11

Count the number of columns that are followed by a larger (i.e. rightwards) column. The &` allows the matches to overlap.

Python 2, 84 bytes

def f(s,k=6):j=(ord(s[0])+('0'<s[0]<':')*2-1)%6;return(j>k)+(~-len(s)and f(s[1:],j))

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Mathematica, 102 bytes

Differences[Last@@Join[Alphabet,ToString/@Range@9,"0"]~Partition~6~Position~#&/@#]~Count~_?(#>0&)&

Pure function. Takes a list of characters as input and returns a number as output. This is a pretty naive solution, golfing suggestions welcome.

C (gcc),  82  79 bytes

o;c(i)i+=i<60&&i>48?1:5;i%=6;f(char*s)for(o=0;*++s;o+=c(*s)>c(*(s-1)));o=o;

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This function will only support lowercase inputs

Ungolfed and commented:

o; //Used for output
c(i) //Calculates the column of given character
 i+= //Correct i to get the correct column
 i<60 //If i is a digit...
 && i>48 //... but not '0'
 ?1 //Then move it one column on the right
 :5; //Else move it five columns on the right
 i%=6; //Get the column number

f(char*s) // The actual "frustrating" function
 for( //Loop for each character
 o=0; //reinitialize output
 *++s; //move to next character / while this is not ''
 o+=c(*s)>c(*(s-1)) //Increment if current character is on the right of the previous one
 );
 o=o; // Outputs result

If my function is allowed to accept wide character strings, it can be reduced to 78 bytes with:

o;c(i)i+=i<60&&i>48?1:5;i%=6;f(int*s)for(o=0;*++s;o+=c(*s)>c(*(s-1)));o=o;

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This version just accept input as int* instead of char*

Edits:

• Golfed 3 bytes in the calculation of the column (function c)

PHP, 74 81 77 bytes

for(;$o=ord($argn[$i]);$i++&&$f+=$p<$x,$p=$x)$x=(--$o/48^1?$o:$o+2)%6;echo$f;

Run as pipe with -nR or try it online.