Here are the (Poor?) translation to the rules of those puzzles. Sorry that I haven't learned Japanese language completely, if there are some people from Japan or someone who is proficient in Japanese language, hopefully they can correct me.
1. Cross sums of 25 July
Fill the grid with digits 1 through 9, making all those equations hold. All those digits must be used exactly once, and all the equations must be always calculated from left to right, from top to bottom, or the basic order of the four operations doesn't apply.
2. Search for "Bulgaria"
Find the words in the word bank (Places in Bulgaria written in Katakana). Not all the letters are used. For the answer entry, submit those unused "letters" from top to bottom.
3. Fill the AKB
Fill the grid with modern English letters, then use lines to connect all the letters to form the names of AKB48 and its counterparts (with "48" omitted). All the letters must be used for exactly one word, and lines can be drawn horizontally, vertically or in 45-degree tilts. For the answer entry, enter the number of words in a straight line (horizontally or vertically), followed by the number of words that form 90-degree turns. Each of the given words can be seen more than once.
Given words: AKB, SKE, NMB, HKT, JKT, SNH, NGT.
4. Paint it black
Shade some cells in the grid, and leave the other cells blank. Numbers outside the grid refer to how many consecutive cells are shaded.
There will be 8 small 5x5 grids to be solved. For the answer entry, identify and enter the small 5x5 grids that belong to the big grid (following the 8 small grids).
5. Pentominoes
Put one of the 12 kinds of pentominoes into the grid. The number of the pentominoes are given outside the grid, and all the pentominoes cannot touch each other, even diagonally. For the answer entry, enter the cells that occupied by pentominoes in the labeled row (In alphabetical order of the labeled rows).
6. Unscramble the celebrities
Using the given clues, restore the letters so that they form a name of a Japanese celebrity.

7. Connected loop
Draw a loop that travels all the cells exactly once. The loop may make a 90-degree turn in either the labeled cell or one of the adjacent neighbouring cells of it. For the answer entry, enter the number of cells that the loop makes 90-degree turns in the respective row.
8. Crossword without clues
Put the words into the blank crossword grid that forms a correct crossword. Words that don't appear in the bank cannot appear in the grid. For the answer entry, enter the contents of the labeled cells.
9. Keep the differences alive
Fill the grid with digits 1 or 2. The consecutive white cells in the same row or column cannot contain the same digit. For the answer entry, enter the contents of the labeled cells.
10. Alphabet connection
Connect the given letters, with a single line attached to the two same letters. All the lines must travel horizontally or vertically, and no two lines cross each other. For the answer entry, enter the given Hiragana letter belong to which English letters' line, respectively.
11. Multiple-choices for missing links
Complete the sequences of letters by choosing one of the answer options. No answers can be chosen more than once. For the answer entry, enter the answers to the tespective labeled sequences.
12. Division of family
Divide the grid into 7 regions of 7 along the borderlines of the cells. Each of the symbols must be entered to either the same or different regions. For the answer entry, enter the number of regions in the respecitive rows.
13. Easy as ABC
Put letters A, B, C, D into some cells of the grid so that all the letters appear exactly once in the same row or column. The shaded diagonal is only used for answer entry. For the answer entry, enter the contents from left to right. Using "X" for blank cells.

14. Division of fields
Divide the grid into L-shaped regions, and their width is 1. All the circles must be always at the turning point of the L-shaped regions, and the regions with the same area cannot share the same edge. For the answer entry, enter the area of region that the labeled cell belong to.
15. Parking lot
Put some 1x2 and 1x3 rectangles (representing cars) into the grid. No two cars overlap each other. Numbers in the cells represent the number of blank cells before and after the car, and it must be occupied by a car. The remaining blank cells must be connected horizontally or vertically. For the answer entry, enter the number of horizontal 1x2 cars, followed by the number of vertical 1x2 ones.
16. Four winds
Draw some lines from the numbered cells so that it travels horizontally or vertically. The numbers inside some cells indicate the total lengths of lines starting from respective cells. No two lines cross each other, and no lines travel through the numbered cells. For the answer entry, enter the number of the labeled cells that the line begin with.
17. Shade it
Shade some cells, leaving the other cells blank. The number and arrow indicates that the number of shaded cells in that direction, and the numbered cells cannot be shaded. For the answer entry, enter the total number of shaded cells.
18. Twin Kakuro
Standard Kakuro rules apply. However, the cell in the same position of the two grids cannot contain the same digit. For the answer entry, enter the contents of the labeled cells.
19. Number block
Put the given numbered blocks into the grid. All the blocks must be put on the edge of the grid (along the outer dotted lines), and the blocks cannot be rotated or reflected. Numbers outside the block indicate the sum of the respective row or column. The cells marked by "X" cannot be occupied by blocks, and the blocks cannot overlap with each other. For the answer entry, enter the sums of the labeled row or column.
20. Nanro
Put the digits 1 to N (N is the size of the region) into the grid so that no two identical digits share the same edge. Also, digits inside the regions must be in a strict ascending or descending order in clockwise or anti-clockwise direction. For the answer entry, enter its contents.